Peregrine falcon broods in the French Jura were surveyed over 40 years (years 1..40). For each year t, the observed number of successful broods C_t and the total number of broods monitored N_t are recorded, together with a standardized year covariate year_t (a real-valued covariate centered and scaled so that year 1 corresponds to approximately -0.95 and each subsequent year increments by 0.05). The intercept alpha has a Normal(mean 0, sd 100) prior. The linear coefficient beta1 on year has a Normal(mean 0, sd 100) prior. The quadratic coefficient beta2 on year-squared has a Normal(mean 0, sd 100) prior.
The log-odds of brood success in year t is a quadratic function of the year covariate: logit_p_t = alpha + beta1 * year_t + beta2 * year_t^2. The observed count of successful broods C_t in year t follows a Binomial distribution with total N_t and success probability inv_logit(logit_p_t).
The marginal posterior distribution of each parameter given the data: the intercept alpha, the linear year coefficient beta1, the quadratic year coefficient beta2, the 40 log-odds values logit_p_1, ..., logit_p_40, and the 40 success probabilities p_1, ..., p_40 (where p_t = inv_logit(logit_p_t)).
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> nyears; // Number of Years3 array[nyears] int<lower=0> C; // Counts4 array[nyears] int<lower=0> N; // Binomial Totals5 vector[nyears] year; // Year covariates6}7transformed data {8 vector[nyears] year_squared;910 year_squared = year .* year;11}12parameters {13 real alpha;14 real beta1;15 real beta2;16}17transformed parameters {18 vector[nyears] logit_p;1920 // Linear predictor21 logit_p = alpha + beta1 * year + beta2 * year_squared;22}23model {24 // Priors25 alpha ~ normal(0, 100);26 beta1 ~ normal(0, 100);27 beta2 ~ normal(0, 100);2829 // Likelihood30 // Distribution for random part31 C ~ binomial_logit(N, logit_p);32}33generated quantities {34 array[nyears] real<lower=0, upper=1> p;3536 for (i in 1 : nyears) {37 p[i] = inv_logit(logit_p[i]);38 }39}4041//@ DATA { nyears: 40, year: [40 values], C: [40 values], N: [40 values] } // values supplied at runtime42//@ PARAMS ["alpha","beta1","beta2","logit_p[1]","logit_p[2]","logit_p[3]","logit_p[4]","logit_p[5]","logit_p[6]","logit_p[7]","logit_p[8]","logit_p[9]","logit_p[10]","logit_p[11]","logit_p[12]","logit_p[13]","logit_p[14]","logit_p[15]","logit_p[16]","logit_p[17]","logit_p[18]","logit_p[19]","logit_p[20]","logit_p[21]","logit_p[22]","logit_p[23]","logit_p[24]","logit_p[25]","logit_p[26]","logit_p[27]","logit_p[28]","logit_p[29]","logit_p[30]","logit_p[31]","logit_p[32]","logit_p[33]","logit_p[34]","logit_p[35]","logit_p[36]","logit_p[37]","logit_p[38]","logit_p[39]","logit_p[40]","p[1]","p[2]","p[3]","p[4]","p[5]","p[6]","p[7]","p[8]","p[9]","p[10]","p[11]","p[12]","p[13]","p[14]","p[15]","p[16]","p[17]","p[18]","p[19]","p[20]","p[21]","p[22]","p[23]","p[24]","p[25]","p[26]","p[27]","p[28]","p[29]","p[30]","p[31]","p[32]","p[33]","p[34]","p[35]","p[36]","p[37]","p[38]","p[39]","p[40]"]43//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}44
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| alpha | 1.01 ± 0.062 | — |
| beta1 | -0.176 ± 0.072 | — |
| beta2 | -0.916 ± 0.148 | — |
| logit_p[1] | 0.355 ± 0.120 | — |
| logit_p[2] | 0.431 ± 0.108 | — |
| logit_p[3] | 0.502 ± 0.097 | — |
| logit_p[4] | 0.569 ± 0.088 | — |
| logit_p[5] | 0.631 ± 0.079 | — |
| logit_p[6] | 0.689 ± 0.072 | — |
| logit_p[7] | 0.742 ± 0.066 | — |
| logit_p[8] | 0.790 ± 0.062 | — |
| logit_p[9] | 0.834 ± 0.059 | — |
| logit_p[10] | 0.873 ± 0.057 | — |
| logit_p[11] | 0.908 ± 0.056 | — |
| logit_p[12] | 0.938 ± 0.056 | — |
| logit_p[13] | 0.964 ± 0.057 | — |
| logit_p[14] | 0.985 ± 0.058 | — |
| logit_p[15] | 1.00 ± 0.059 | — |
| logit_p[16] | 1.01 ± 0.060 | — |
| logit_p[17] | 1.02 ± 0.061 | — |
| logit_p[18] | 1.02 ± 0.062 | — |
| logit_p[19] | 1.02 ± 0.062 | — |
| logit_p[20] | 1.01 ± 0.062 | — |
| logit_p[21] | 1.00 ± 0.062 | — |
| logit_p[22] | 0.988 ± 0.061 | — |
| logit_p[23] | 0.967 ± 0.060 | — |
| logit_p[24] | 0.942 ± 0.059 | — |
| logit_p[25] | 0.913 ± 0.058 | — |
| logit_p[26] | 0.879 ± 0.057 | — |
| logit_p[27] | 0.840 ± 0.055 | — |
| logit_p[28] | 0.797 ± 0.054 | — |
| logit_p[29] | 0.750 ± 0.054 | — |
| logit_p[30] | 0.697 ± 0.055 | — |
| logit_p[31] | 0.640 ± 0.056 | — |
| logit_p[32] | 0.579 ± 0.059 | — |
| logit_p[33] | 0.513 ± 0.063 | — |
| logit_p[34] | 0.442 ± 0.069 | — |
| logit_p[35] | 0.367 ± 0.076 | — |
| logit_p[36] | 0.287 ± 0.085 | — |
| logit_p[37] | 0.203 ± 0.094 | — |
| logit_p[38] | 0.114 ± 0.105 | — |
| logit_p[39] | 0.020 ± 0.117 | — |
| logit_p[40] | -0.078 ± 0.130 | — |
| p[1] | 0.587 ± 0.029 | — |
| p[2] | 0.606 ± 0.026 | — |
| p[3] | 0.623 ± 0.023 | — |
| p[4] | 0.638 ± 0.020 | — |
| p[5] | 0.653 ± 0.018 | — |
| p[6] | 0.665 ± 0.016 | — |
| p[7] | 0.677 ± 0.014 | — |
| p[8] | 0.688 ± 0.013 | — |
| p[9] | 0.697 ± 0.012 | — |
| p[10] | 0.705 ± 0.012 | — |
| p[11] | 0.712 ± 0.012 | — |
| p[12] | 0.719 ± 0.011 | — |
| p[13] | 0.724 ± 0.011 | — |
| p[14] | 0.728 ± 0.011 | — |
| p[15] | 0.731 ± 0.012 | — |
| p[16] | 0.733 ± 0.012 | — |
| p[17] | 0.735 ± 0.012 | — |
| p[18] | 0.735 ± 0.012 | — |
| p[19] | 0.735 ± 0.012 | — |
| p[20] | 0.734 ± 0.012 | — |
| p[21] | 0.732 ± 0.012 | — |
| p[22] | 0.728 ± 0.012 | — |
| p[23] | 0.724 ± 0.012 | — |
| p[24] | 0.719 ± 0.012 | — |
| p[25] | 0.713 ± 0.012 | — |
| p[26] | 0.707 ± 0.012 | — |
| p[27] | 0.698 ± 0.012 | — |
| p[28] | 0.689 ± 0.012 | — |
| p[29] | 0.679 ± 0.012 | — |
| p[30] | 0.667 ± 0.012 | — |
| p[31] | 0.655 ± 0.013 | — |
| p[32] | 0.641 ± 0.014 | — |
| p[33] | 0.625 ± 0.015 | — |
| p[34] | 0.609 ± 0.016 | — |
| p[35] | 0.591 ± 0.018 | — |
| p[36] | 0.571 ± 0.021 | — |
| p[37] | 0.550 ± 0.023 | — |
| p[38] | 0.528 ± 0.026 | — |
| p[39] | 0.505 ± 0.029 | — |
| p[40] | 0.481 ± 0.032 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0056 ≤ tol 0.0189 · floors 0.0095/0.0089 |
Peregrine falcon breeding-pair counts C_t and standardized year values year_t are observed for n = 40 consecutive years (t = 1..40); the counts C_t and the year values year_t are provided as data arrays. The intercept alpha has a Uniform(-20, 20) prior. The linear-year coefficient beta1 has a Uniform(-10, 10) prior. The quadratic-year coefficient beta2 has a Uniform(-10, 10) prior. The cubic-year coefficient beta3 has a Uniform(-10, 10) prior.
The log expected count in year t is a cubic polynomial in year_t: log_lambda_t = alpha + beta1 * year_t + beta2 * year_t^2 + beta3 * year_t^3. The observed count C_t is drawn from a Poisson distribution with mean lambda_t = exp(log_lambda_t), independently across years.
The marginal posterior distribution of each parameter and derived quantity given the data: the intercept alpha, the three polynomial coefficients beta1, beta2, beta3, the 40 log-scale expected counts log_lambda_1, ..., log_lambda_40, and the 40 expected counts on the original scale lambda_1, ..., lambda_40.
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> n; // Number of years3 array[n] int<lower=0> C; // Count4 vector[n] year; // Year5}6transformed data {7 vector[n] year_squared;8 vector[n] year_cubed;910 year_squared = square(year);11 year_cubed = year_squared .* year;12}13parameters {14 real<lower=-20, upper=20> alpha;15 real<lower=-10, upper=10> beta1;16 real<lower=-10, upper=10> beta2;17 real<lower=-10, upper=10> beta3;18}19transformed parameters {20 vector[n] log_lambda;2122 log_lambda = alpha + beta1 * year + +beta2 * year_squared23 + +beta3 * year_cubed;24}25model {26 // Implicit uniform priors are used.2728 // Likelihood29 C ~ poisson_log(log_lambda);30}31generated quantities {32 vector[n] lambda;3334 lambda = exp(log_lambda);35}3637//@ DATA { year: [40 values], C: [40 values], n: 40 } // values supplied at runtime38//@ PARAMS ["alpha","beta1","beta2","beta3","log_lambda[1]","log_lambda[2]","log_lambda[3]","log_lambda[4]","log_lambda[5]","log_lambda[6]","log_lambda[7]","log_lambda[8]","log_lambda[9]","log_lambda[10]","log_lambda[11]","log_lambda[12]","log_lambda[13]","log_lambda[14]","log_lambda[15]","log_lambda[16]","log_lambda[17]","log_lambda[18]","log_lambda[19]","log_lambda[20]","log_lambda[21]","log_lambda[22]","log_lambda[23]","log_lambda[24]","log_lambda[25]","log_lambda[26]","log_lambda[27]","log_lambda[28]","log_lambda[29]","log_lambda[30]","log_lambda[31]","log_lambda[32]","log_lambda[33]","log_lambda[34]","log_lambda[35]","log_lambda[36]","log_lambda[37]","log_lambda[38]","log_lambda[39]","log_lambda[40]","lambda[1]","lambda[2]","lambda[3]","lambda[4]","lambda[5]","lambda[6]","lambda[7]","lambda[8]","lambda[9]","lambda[10]","lambda[11]","lambda[12]","lambda[13]","lambda[14]","lambda[15]","lambda[16]","lambda[17]","lambda[18]","lambda[19]","lambda[20]","lambda[21]","lambda[22]","lambda[23]","lambda[24]","lambda[25]","lambda[26]","lambda[27]","lambda[28]","lambda[29]","lambda[30]","lambda[31]","lambda[32]","lambda[33]","lambda[34]","lambda[35]","lambda[36]","lambda[37]","lambda[38]","lambda[39]","lambda[40]"]39//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}40
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| alpha | 4.28 ± 0.029 | — |
| beta1 | 1.25 ± 0.045 | — |
| beta2 | 0.069 ± 0.024 | — |
| beta3 | -0.229 ± 0.024 | — |
| log_lambda[1] | 3.46 ± 0.102 | — |
| log_lambda[2] | 3.39 ± 0.087 | — |
| log_lambda[3] | 3.34 ± 0.074 | — |
| log_lambda[4] | 3.31 ± 0.064 | — |
| log_lambda[5] | 3.29 ± 0.055 | — |
| log_lambda[6] | 3.28 ± 0.049 | — |
| log_lambda[7] | 3.29 ± 0.045 | — |
| log_lambda[8] | 3.31 ± 0.043 | — |
| log_lambda[9] | 3.34 ± 0.042 | — |
| log_lambda[10] | 3.39 ± 0.041 | — |
| log_lambda[11] | 3.44 ± 0.041 | — |
| log_lambda[12] | 3.50 ± 0.041 | — |
| log_lambda[13] | 3.57 ± 0.040 | — |
| log_lambda[14] | 3.65 ± 0.040 | — |
| log_lambda[15] | 3.74 ± 0.039 | — |
| log_lambda[16] | 3.83 ± 0.037 | — |
| log_lambda[17] | 3.92 ± 0.036 | — |
| log_lambda[18] | 4.02 ± 0.034 | — |
| log_lambda[19] | 4.13 ± 0.032 | — |
| log_lambda[20] | 4.23 ± 0.030 | — |
| log_lambda[21] | 4.34 ± 0.028 | — |
| log_lambda[22] | 4.44 ± 0.026 | — |
| log_lambda[23] | 4.55 ± 0.024 | — |
| log_lambda[24] | 4.66 ± 0.023 | — |
| log_lambda[25] | 4.76 ± 0.023 | — |
| log_lambda[26] | 4.86 ± 0.022 | — |
| log_lambda[27] | 4.96 ± 0.022 | — |
| log_lambda[28] | 5.05 ± 0.022 | — |
| log_lambda[29] | 5.14 ± 0.023 | — |
| log_lambda[30] | 5.22 ± 0.023 | — |
| log_lambda[31] | 5.29 ± 0.023 | — |
| log_lambda[32] | 5.36 ± 0.022 | — |
| log_lambda[33] | 5.41 ± 0.022 | — |
| log_lambda[34] | 5.46 ± 0.021 | — |
| log_lambda[35] | 5.50 ± 0.020 | — |
| log_lambda[36] | 5.52 ± 0.020 | — |
| log_lambda[37] | 5.54 ± 0.022 | — |
| log_lambda[38] | 5.53 ± 0.026 | — |
| log_lambda[39] | 5.52 ± 0.032 | — |
| log_lambda[40] | 5.49 ± 0.040 | — |
| lambda[1] | 32.03 ± 3.27 | — |
| lambda[2] | 29.88 ± 2.60 | — |
| lambda[3] | 28.36 ± 2.10 | — |
| lambda[4] | 27.36 ± 1.74 | — |
| lambda[5] | 26.82 ± 1.49 | — |
| lambda[6] | 26.66 ± 1.32 | — |
| lambda[7] | 26.87 ± 1.22 | — |
| lambda[8] | 27.43 ± 1.18 | — |
| lambda[9] | 28.33 ± 1.19 | — |
| lambda[10] | 29.59 ± 1.22 | — |
| lambda[11] | 31.21 ± 1.28 | — |
| lambda[12] | 33.23 ± 1.36 | — |
| lambda[13] | 35.67 ± 1.44 | — |
| lambda[14] | 38.58 ± 1.53 | — |
| lambda[15] | 42.00 ± 1.63 | — |
| lambda[16] | 45.98 ± 1.72 | — |
| lambda[17] | 50.60 ± 1.81 | — |
| lambda[18] | 55.89 ± 1.89 | — |
| lambda[19] | 61.94 ± 1.97 | — |
| lambda[20] | 68.80 ± 2.05 | — |
| lambda[21] | 76.53 ± 2.13 | — |
| lambda[22] | 85.18 ± 2.22 | — |
| lambda[23] | 94.79 ± 2.32 | — |
| lambda[24] | 105.4 ± 2.46 | — |
| lambda[25] | 116.9 ± 2.64 | — |
| lambda[26] | 129.3 ± 2.88 | — |
| lambda[27] | 142.4 ± 3.16 | — |
| lambda[28] | 156.2 ± 3.49 | — |
| lambda[29] | 170.5 ± 3.84 | — |
| lambda[30] | 184.8 ± 4.18 | — |
| lambda[31] | 198.9 ± 4.47 | — |
| lambda[32] | 212.4 ± 4.70 | — |
| lambda[33] | 224.8 ± 4.84 | — |
| lambda[34] | 235.6 ± 4.90 | — |
| lambda[35] | 244.3 ± 4.95 | — |
| lambda[36] | 250.4 ± 5.12 | — |
| lambda[37] | 253.6 ± 5.59 | — |
| lambda[38] | 253.4 ± 6.50 | — |
| lambda[39] | 249.7 ± 7.85 | — |
| lambda[40] | 242.5 ± 9.56 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.3618 ≤ tol 1.4472 · floors 0.3962/0.3972 |
A capture-recapture study was conducted over T = 3 sampling occasions. Data augmentation was applied: the observed capture histories are embedded in an augmented dataset of M = 237 individuals, where each individual i has a binary capture history y[i,1..T] (1 = captured, 0 = not captured on that occasion). Individuals with all zeros (never captured) are included in the augmented portion. The inclusion probability omega and the detection probability p both have implicit Uniform(0, 1) priors.
Each of the M candidate individuals in the augmented dataset is either a real member of the population (with probability omega) or a phantom (with probability 1 - omega). If an individual is real, it is detected independently on each of the T occasions with probability p. If it is a phantom, it is never detected. The discrete membership indicator for each individual is marginalized out: an individual with at least one capture is certainly real; an individual never captured is real (but missed on all T occasions) with probability omega * (1-p)^T, or is a phantom with probability 1 - omega.
The marginal posterior distribution of each parameter given the data: the inclusion probability omega, the detection probability p, the probability of being present given never detected omega_nd (= omega * (1-p)^T / (omega * (1-p)^T + (1-omega))), and the estimated total population size N (= number observed + Binomial(M - number observed, omega_nd) draw).
answer spec
{
"kind": "record",
"fields": {
"omega": {
"kind": "dist",
"domain": "real"
},
"p": {
"kind": "dist",
"domain": "real"
},
"omega_nd": {
"kind": "dist",
"domain": "real"
},
"N": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> M; // Size of augumented data set3 int<lower=0> T; // Number of sampling occasions4 array[M, T] int<lower=0, upper=1> y; // Capture-history matrix5}6transformed data {7 array[M] int<lower=0> s; // Totals in each row8 int<lower=0> C; // Size of observed data set910 C = 0;11 for (i in 1 : M) {12 s[i] = sum(y[i]);13 if (s[i] > 0) {14 C = C + 1;15 }16 }17}18parameters {19 real<lower=0, upper=1> omega; // Inclusion probability20 real<lower=0, upper=1> p; // Detection probability21}22model {23 // Priors are imlicitly defined;24 // omega ~ uniform(0, 1);25 // p ~ uniform(0, 1);2627 // Likelihood28 for (i in 1 : M) {29 if (s[i] > 0) {30 // z[i] == 131 target += bernoulli_lpmf(1 | omega) + binomial_lpmf(s[i] | T, p);32 } else // s[i] == 033 {34 target += log_sum_exp(bernoulli_lpmf(1 | omega)35 // z[i] == 136 + binomial_lpmf(0 | T, p),37 bernoulli_lpmf(0 | omega));38 }39 } // z[i] == 040}41generated quantities {42 // prob present given never detected43 real omega_nd = (omega * (1 - p) ^ T) / (omega * (1 - p) ^ T + (1 - omega));44 int<lower=C, upper=M> N = C + binomial_rng(M - C, omega_nd);45}4647//@ DATA { y: [237×3 matrix], M: 237, T: 3 } // values supplied at runtime48//@ PARAMS ["omega","p","omega_nd","N"]49//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}50
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.1598 ≤ tol 0.8445 · floors 0.4223/0.2935 |
A capture-recapture study was conducted over T = 3 sampling occasions, with detection probabilities that may differ by occasion. Data augmentation was applied: the observed capture histories are embedded in an augmented dataset of M = 237 individuals, where each individual i has a binary capture history y[i,1..T]. The inclusion probability omega has an implicit Uniform(0, 1) prior. Each occasion-specific detection probability p_t (t = 1, 2, 3) has an implicit Uniform(0, 1) prior, independently.
Each of the M candidate individuals is either a real member of the population (with probability omega) or a phantom (with probability 1 - omega). If an individual is real, it is detected on occasion t independently with probability p_t, which may vary by occasion. If it is a phantom, it is never detected. The membership indicator is marginalized out as in the constant-p model, but now using the occasion-specific p_t values.
The marginal posterior distribution of each parameter given the data: the inclusion probability omega, the three occasion-specific detection probabilities p_1, p_2, p_3, the probability of never being captured given presence pr (= product of (1 - p_t) over all T occasions), the probability of being present given never detected omega_nd (= omega * pr / (omega * pr + (1 - omega))), and the estimated total population size N.
answer spec
{
"kind": "record",
"fields": {
"omega": {
"kind": "dist",
"domain": "real"
},
"p[1]": {
"kind": "dist",
"domain": "real"
},
"p[2]": {
"kind": "dist",
"domain": "real"
},
"p[3]": {
"kind": "dist",
"domain": "real"
},
"pr": {
"kind": "dist",
"domain": "real"
},
"omega_nd": {
"kind": "dist",
"domain": "real"
},
"N": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> M; // Size of augumented data set3 int<lower=0> T; // Number of sampling occasions4 array[M, T] int<lower=0, upper=1> y; // Capture-history matrix5}6transformed data {7 array[M] int<lower=0> s; // Totals in each row8 int<lower=0> C; // Size of observed data set910 C = 0;11 for (i in 1 : M) {12 s[i] = sum(y[i]);13 if (s[i] > 0) {14 C = C + 1;15 }16 }17}18parameters {19 real<lower=0, upper=1> omega; // Inclusion probability20 vector<lower=0, upper=1>[T] p; // Detection probability21}22model {23 // Priors are implicitly defined.24 // omega ~ uniform(0, 1);25 // p ~ uniform(0, 1);2627 // Likelihood28 for (i in 1 : M) {29 if (s[i] > 0) {30 // z[i] == 131 target += bernoulli_lpmf(1 | omega) + bernoulli_lpmf(y[i] | p);32 } else // s[i] == 033 {34 target += log_sum_exp(bernoulli_lpmf(1 | omega)35 // z[i] == 136 + bernoulli_lpmf(y[i] | p),37 bernoulli_lpmf(0 | omega));38 }39 } // z[i] == 040}41generated quantities {42 // prob never captured given present43 real pr = prod(1 - p);44 // prob present given never captured; same for all animals45 real omega_nd = (omega * pr) / (omega * pr + (1 - omega));46 int<lower=C> N = C + binomial_rng(M - C, omega_nd);47}4849//@ DATA { y: [237×3 matrix], M: 237, T: 3 } // values supplied at runtime50//@ PARAMS ["omega","p[1]","p[2]","p[3]","pr","omega_nd","N"]51//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}52
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.1708 ≤ tol 0.6700 · floors 0.2843/0.3230 |
A subject answered k = 5 questions correctly out of n = 10 questions of equal difficulty. The unknown success rate theta is constrained to lie in [0, 1] and has a Beta(1, 1) prior.
Each question is answered correctly or incorrectly with the same probability theta, independently of all other questions. The total number of correct answers follows a Binomial distribution with count n and success probability theta.
The marginal posterior distribution of the success rate theta given the data.
answer spec
{
"kind": "record",
"fields": {
"theta": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// Inferring a Rate2data {3 int<lower=1> n;4 int<lower=0> k;5}6parameters {7 real<lower=0, upper=1> theta;8}9model {10 // Prior Distribution for Rate Theta11 theta ~ beta(1, 1);1213 // Observed Counts14 k ~ binomial(n, theta);15}1617//@ DATA { k: 5, n: 10 } // values supplied at runtime18//@ PARAMS ["theta"]19//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}20
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0041 ≤ tol 0.0164 · floors 0.0082/0.0049 |
Two independent binomial trials were conducted. Trial 1 had n1 = 10 attempts with k1 = 5 successes. Trial 2 had n2 = 10 attempts with k2 = 7 successes. The success rate theta1 for trial 1 has a Beta(1, 1) prior (i.e., Uniform(0, 1)). The success rate theta2 for trial 2 has a Beta(1, 1) prior (i.e., Uniform(0, 1)), independently.
The number of successes in trial 1 is Binomial with n1 attempts and success probability theta1. The number of successes in trial 2 is Binomial with n2 attempts and success probability theta2. The difference delta = theta1 - theta2 is a derived quantity constrained to lie in (-1, 1).
The marginal posterior distribution of each parameter given the data: the success rate theta1, the success rate theta2, and the difference delta = theta1 - theta2.
answer spec
{
"kind": "record",
"fields": {
"theta1": {
"kind": "dist",
"domain": "real"
},
"theta2": {
"kind": "dist",
"domain": "real"
},
"delta": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// Difference Between Two Rates2data {3 int<lower=1> n1;4 int<lower=1> n2;5 int<lower=0> k1;6 int<lower=0> k2;7}8parameters {9 real<lower=0, upper=1> theta1;10 real<lower=0, upper=1> theta2;11}12transformed parameters {13 real<lower=-1, upper=1> delta;14 delta = theta1 - theta2;15}16model {17 // Prior Distribution for Rate Theta18 theta1 ~ beta(1, 1);19 theta2 ~ beta(1, 1);20 // Observed Counts21 k1 ~ binomial(n1, theta1);22 k2 ~ binomial(n2, theta2);23}2425//@ DATA { n1: 10, n2: 10, k1: 5, k2: 7 } // values supplied at runtime26//@ PARAMS ["theta1","theta2","delta"]27//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}28
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0051 ≤ tol 0.0361 · floors 0.0077/0.0082 |
Two independent binomial trials are assumed to share a common success rate. Trial 1 had n1 = 10 attempts with k1 = 5 successes. Trial 2 had n2 = 10 attempts with k2 = 7 successes. The common success rate theta has a Beta(1, 1) prior (i.e., Uniform(0, 1)).
Both trials share the same underlying success probability theta. The number of successes in trial 1 is Binomial with n1 attempts and probability theta. The number of successes in trial 2 is Binomial with n2 attempts and probability theta, independently of trial 1.
The marginal posterior distribution of theta given the data.
answer spec
{
"kind": "record",
"fields": {
"theta": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// // Inferring a Common Rate2data {3 int<lower=1> n1;4 int<lower=1> n2;5 int<lower=0> k1;6 int<lower=0> k2;7}8parameters {9 real<lower=0, upper=1> theta;10}11model {12 // Prior on Single Rate Theta13 theta ~ beta(1, 1);14 // Observed Counts15 k1 ~ binomial(n1, theta);16 k2 ~ binomial(n2, theta);17}1819//@ DATA { n1: 10, n2: 10, k1: 5, k2: 7 } // values supplied at runtime20//@ PARAMS ["theta"]21//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}22
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0056 ≤ tol 0.0137 · floors 0.0068/0.0066 |
A binomial trial had n = 15 attempts with k = 1 success. Two parameters are modeled: the posterior rate theta (conditioned on the data) and an independent prior rate thetaprior (not conditioned on the data, serving as a prior predictive auxiliary). Both theta and thetaprior have Beta(1, 1) priors, independently.
The observed count k is Binomial with n attempts and success probability theta. The parameter thetaprior is drawn from the same Beta(1, 1) prior but is not updated by any data — it remains a prior draw.
The marginal posterior distribution of each parameter given the data: the posterior rate theta, the independent prior rate thetaprior, the posterior predictive count postpredk (a Binomial(n, theta) draw), and the prior predictive count priorpredk (a Binomial(n, thetaprior) draw).
answer spec
{
"kind": "record",
"fields": {
"theta": {
"kind": "dist",
"domain": "real"
},
"thetaprior": {
"kind": "dist",
"domain": "real"
},
"postpredk": {
"kind": "dist",
"domain": "real"
},
"priorpredk": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// Prior and Posterior Prediction2data {3 int<lower=1> n;4 int<lower=0> k;5}6parameters {7 real<lower=0, upper=1> theta;8 real<lower=0, upper=1> thetaprior;9}10model {11 // Prior on Rate Theta12 theta ~ beta(1, 1);13 thetaprior ~ beta(1, 1);14 // Observed Data15 k ~ binomial(n, theta);16}17generated quantities {18 int<lower=0> postpredk;19 int<lower=0> priorpredk;2021 // Posterior Predictive22 postpredk = binomial_rng(n, theta);23 // Prior Predictive24 priorpredk = binomial_rng(n, thetaprior);25}2627//@ DATA { k: 1, n: 15 } // values supplied at runtime28//@ PARAMS ["theta","thetaprior","postpredk","priorpredk"]29//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}30
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.1365 ≤ tol 0.8740 · floors 0.2003/0.2570 |
Two independent binomial trials are assumed to share a common success rate. Trial 1 had n1 = 10 attempts with k1 = 0 successes. Trial 2 had n2 = 10 attempts with k2 = 10 successes. The common success rate theta has a Beta(1, 1) prior (i.e., Uniform(0, 1)).
Both trials share the same underlying success probability theta. The number of successes in trial 1 is Binomial with n1 attempts and probability theta. The number of successes in trial 2 is Binomial with n2 attempts and probability theta, independently of trial 1.
The marginal posterior distribution of each parameter given the data: the common success rate theta, the posterior predictive count postpredk1 for a new trial of size n1 (a Binomial(n1, theta) draw), and the posterior predictive count postpredk2 for a new trial of size n2 (a Binomial(n2, theta) draw).
answer spec
{
"kind": "record",
"fields": {
"theta": {
"kind": "dist",
"domain": "real"
},
"postpredk1": {
"kind": "dist",
"domain": "real"
},
"postpredk2": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// Inferring a Common Rate, With Posterior Predictive2data {3 int<lower=1> n1;4 int<lower=1> n2;5 int<lower=0> k1;6 int<lower=0> k2;7}8parameters {9 real<lower=0, upper=1> theta;10}11model {12 // Prior on Single Rate Theta13 theta ~ beta(1, 1);1415 // Observed Counts16 k1 ~ binomial(n1, theta);17 k2 ~ binomial(n2, theta);18}19generated quantities {20 int<lower=0, upper=n1> postpredk1;21 int<lower=0, upper=n2> postpredk2;2223 // Posterior Predictive24 postpredk1 = binomial_rng(n1, theta);25 postpredk2 = binomial_rng(n2, theta);26}2728//@ DATA { n1: 10, n2: 10, k1: 0, k2: 10 } // values supplied at runtime29//@ PARAMS ["theta","postpredk1","postpredk2"]30//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}31
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0517 ≤ tol 0.3290 · floors 0.0890/0.1645 |
A number of surveys were sent out and some were returned; the returns are observed on m = 5 separate occasions, giving five observed return counts (each a non-negative integer). The maximum plausible number of surveys sent is 500. The unknown number of surveys sent is at most 500 and at least the largest observed return count, with a discrete uniform prior over that range. The return rate theta has a Uniform(0, 1) prior.
A fixed but unknown number of surveys n was sent, and each survey is independently returned with probability theta. On each observation occasion the number of returns is Binomial with n trials and success probability theta. The unknown count n is marginalized over its full range (from the largest observed return count up to 500), so the return rate theta is inferred by averaging over all consistent values of n.
The marginal posterior distribution of the return rate theta given the data.
answer spec
{
"kind": "record",
"fields": {
"theta": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1/*2 * Notes to Stan model3 * --------------------4 * This model code is more difficult to understand in Stan implementation since5 * Stan is unable to sample discrete parameters. This may change in the future.6 * For better understanding read Stan manual chapter 'Mixture Modeling' (p.68 in7 * version 2.4.0) first.8 */910// Inferring Return Rate and Number of Surveys from Observed Returns11data {12 int<lower=0> nmax;13 int<lower=0> m;14 array[m] int<lower=0, upper=nmax> k;15}16transformed data {17 int<lower=0> nmin; // Minimal possible n1819 nmin = max(k);20}21parameters {22 real<lower=0, upper=1> theta;23}24transformed parameters {25 vector[nmax] lp_parts; // Log probability for each n2627 // First part of the trick for mixture model28 for (n in 1 : nmax) {29 if (n < nmin) {30 lp_parts[n] = log(1.0 / nmax) + negative_infinity();31 } // Zero probability32 else {33 lp_parts[n] = log(1.0 / nmax) + binomial_lpmf(k | n, theta);34 }35 }36 // Deprecated: lp_parts[n] = log(1.0 / nmax) + binomial_log(k, n, theta);37}38model {39 // Second part of the trick for mixture model40 // Deprecated: increment_log_prob(log_sum_exp(lp_parts));41 target += log_sum_exp(lp_parts);42}43generated quantities {44 int<lower=1, upper=nmax> n;45 simplex[nmax] prob_n;4647 // Transforming lp_parts to probabilities of each n48 prob_n = softmax(lp_parts);49 n = categorical_rng(prob_n);50}5152//@ DATA { nmax: 500, k: [5 values], m: 5 } // values supplied at runtime53//@ PARAMS ["theta"]54//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}55
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0050 ≤ tol 0.0298 · floors 0.0149/0.0047 |
For a time series of T observations, the data provide an array y of T real-valued observations. The model also requires K, the number of autoregressive lags, which is fixed. The intercept parameter alpha has a Normal(location 0, scale 10) prior. Each of the K autoregressive lag coefficients beta[1] through beta[K] has a Normal(location 0, scale 10) prior. The observation noise standard deviation sigma, constrained to be positive, has a half-Cauchy(location 0, scale 2.5) prior.
An autoregressive time series model of order K. The first K observations serve as initial conditions. For each subsequent time point t from K+1 to T, the observation y[t] is generated from a normal distribution whose mean is the intercept alpha plus a linear autoregressive term (the sum of beta[k] times the observation from k steps in the past, y[t-k], for k from 1 to K), and whose standard deviation is sigma.
The marginal posterior distributions of the following parameters: alpha (the intercept), beta[1] through beta[K] (the K autoregressive lag coefficients), and sigma (the observation noise standard deviation).
answer spec
{
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"alpha": {
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"beta[1]": {
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}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> K;3 int<lower=0> T;4 array[T] real y;5}6parameters {7 real alpha;8 array[K] real beta;9 real<lower=0> sigma;10}11model {12 alpha ~ normal(0, 10);13 beta ~ normal(0, 10);14 sigma ~ cauchy(0, 2.5);1516 for (t in (K + 1) : T) {17 real mu;18 mu = alpha;1920 for (k in 1 : K) {21 mu = mu + beta[k] * y[t - k];22 }2324 y[t] ~ normal(mu, sigma);25 }26}2728//@ DATA { K: 5, T: 200, y: [200 values] } // values supplied at runtime29//@ PARAMS ["alpha","beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","sigma"]30//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}31
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0038 ≤ tol 0.0132 · floors 0.0063/0.0066 |
For a sequence of N = 416 observations of basketball drive events, the data provide two measurements at each time step: u[t], the inverse player speed (1/speed), and v[t], the distance to the hoop. The model operates with K = 2 hidden states (state 1 represents no drive event, state 2 represents a drive event). A matrix alpha, of size 2 by 2 with positive entries, supplies the Dirichlet hyperparameters for the transition probability priors; each row k specifies the hyperparameters for transitions originating from state k. The parameters theta1 and theta2 are probability vectors that sum to 1 (simplices), each with a Dirichlet prior; theta1 governs transitions from state 1 and theta2 governs transitions from state 2, with hyperparameters from alpha[1, :] and alpha[2, :] respectively. The parameters phi and lambda are pairs of positive values constrained so that phi[1] <= phi[2] and lambda[1] <= lambda[2]; phi[1] has a Normal(0, 1) prior, phi[2] has a Normal(3, 1) prior, lambda[1] has a Normal(0, 1) prior, and lambda[2] has a Normal(3, 1) prior. All four of these priors are over the positive reals (half-normal distributions).
The observed sequence is generated by a discrete-time hidden Markov model with K = 2 hidden states. At the initial time step t = 1, the hidden state z[1] is drawn uniformly at random from the two states. At each subsequent time step t = 2, 3, ..., N, the hidden state z[t] transitions from z[t-1] according to a state-specific transition probability vector: if z[t-1] = 1, the next state is drawn from the categorical distribution determined by theta1; if z[t-1] = 2, the next state is drawn from theta2. Given the hidden state z[t] at time t, the two observations u[t] and v[t] are generated independently as exponential random variables with rates phi[z[t]] and lambda[z[t]] respectively.
The marginal posterior distributions of the eight parameters: theta1[1] and theta1[2] (the state-conditional transition probabilities from state 1 to states 1 and 2 respectively), theta2[1] and theta2[2] (the state-conditional transition probabilities from state 2 to states 1 and 2 respectively), phi[1] and phi[2] (the exponential rate parameters for inverse speed in states 1 and 2), and lambda[1] and lambda[2] (the exponential rate parameters for hoop distance in states 1 and 2).
answer spec
{
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// drive model (exponential dist)2data {3 int<lower=1> K; // number of states (1 = none, 2 = drive)4 int<lower=1> N; // length of process5 array[N] real u; // 1/speed6 array[N] real v; // hoop distance7 matrix<lower=0>[K, K] alpha; // transit prior8}9parameters {10 simplex[K] theta1;11 simplex[K] theta2;12 // enforce an ordering: phi[1] <= phi[2]13 positive_ordered[K] phi; // emission parameter for 1/speed14 positive_ordered[K] lambda; // emission parameter for hoop dist15}16transformed parameters {17 array[K] simplex[K] theta; // transit probs18 theta[1] = theta1;19 theta[2] = theta2;20}21model {22 // priors23 for (k in 1 : K) {24 target += dirichlet_lpdf(theta[k] | alpha[k, : ]');25 }26 target += normal_lpdf(phi[1] | 0, 1);27 target += normal_lpdf(phi[2] | 3, 1);28 target += normal_lpdf(lambda[1] | 0, 1);29 target += normal_lpdf(lambda[2] | 3, 1);30 // forward algorithm31 {32 array[K] real acc;33 array[N, K] real gamma;34 for (k in 1 : K) {35 gamma[1, k] = exponential_lpdf(u[1] | phi[k])36 + exponential_lpdf(v[1] | lambda[k]);37 }38 for (t in 2 : N) {39 for (k in 1 : K) {40 for (j in 1 : K) {41 acc[j] = gamma[t - 1, j] + log(theta[j, k])42 + exponential_lpdf(u[t] | phi[k])43 + exponential_lpdf(v[t] | lambda[k]);44 }45 gamma[t, k] = log_sum_exp(acc);46 }47 }48 target += log_sum_exp(gamma[N]);49 }50}51generated quantities {52 array[N] int<lower=1, upper=K> z_star;53 real log_p_z_star;54 // Viterbi algorithm55 {56 array[N, K] int back_ptr;57 array[N, K] real best_logp;58 for (k in 1 : K) {59 best_logp[1, K] = exponential_lpdf(u[1] | phi[k])60 + exponential_lpdf(v[1] | lambda[k]);61 }62 for (t in 2 : N) {63 for (k in 1 : K) {64 best_logp[t, k] = negative_infinity();65 for (j in 1 : K) {66 real logp;67 logp = best_logp[t - 1, j] + log(theta[j, k])68 + exponential_lpdf(u[t] | phi[k])69 + exponential_lpdf(v[t] | lambda[k]);70 if (logp > best_logp[t, k]) {71 back_ptr[t, k] = j;72 best_logp[t, k] = logp;73 }74 }75 }76 }77 log_p_z_star = max(best_logp[N]);78 for (k in 1 : K) {79 if (best_logp[N, k] == log_p_z_star) {80 z_star[N] = k;81 }82 }83 for (t in 1 : (N - 1)) {84 z_star[N - t] = back_ptr[N - t + 1, z_star[N - t + 1]];85 }86 }87}8889//@ DATA { N: 416, K: 2, u: [416 values], v: [416 values], alpha: [2×2 matrix] } // values supplied at runtime90//@ PARAMS ["theta1[1]","theta1[2]","theta2[1]","theta2[2]","phi[1]","phi[2]","lambda[1]","lambda[2]"]91//@ SAMPLING {"chains":8,"iter_warmup":6000,"iter_sampling":3000,"adapt_delta":0.9}92
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0177 ≤ tol 0.0727 · floors 0.0080/0.0307 |
For each of N time steps the data provide two real-valued observations: u[t], a movement measurement (inverse speed, which can be negative), and v[t], the distance to the hoop. The hidden Markov model has K = 2 states (state 1 = no drive, state 2 = drive). Transitions out of each state are governed by a transition probability vector — theta1 out of state 1 and theta2 out of state 2 — each a probability vector over the K states that sums to 1, with a Dirichlet prior whose concentration parameters are supplied as the rows of a K-by-K positive matrix alpha (row k for transitions out of state k). The per-state emission means are phi[1], phi[2] for u and lambda[1], lambda[2] for v; phi and lambda are each ordered so that phi[1] <= phi[2] and lambda[1] <= lambda[2]. The priors are phi[1] ~ Normal(0, 1), phi[2] ~ Normal(3, 1), lambda[1] ~ Normal(0, 1), and lambda[2] ~ Normal(3, 1). The emission standard deviations are fixed and supplied as data: tau for u and rho for v.
The observed sequence is generated by a two-state hidden Markov model. The hidden state at the first time step is equally likely to be either state. At each subsequent step the hidden state transitions from the previous one according to that state's transition vector (theta1 if the previous state was 1, theta2 if it was 2). Given the hidden state k at a time step, the two observations are generated independently and normally: u[t] ~ Normal(phi[k], tau) and v[t] ~ Normal(lambda[k], rho).
The marginal posterior distribution of each of the eight parameters: theta1[1], theta1[2] (the probabilities of transitioning from state 1 to states 1 and 2), theta2[1], theta2[2] (from state 2 to states 1 and 2), phi[1], phi[2] (the u-emission means for states 1 and 2, with phi[1] <= phi[2]), and lambda[1], lambda[2] (the v-emission means for states 1 and 2, with lambda[1] <= lambda[2]).
answer spec
{
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// drive model (normal dist)2data {3 int<lower=1> K; // number of states (1 = none, 2 = drive)4 int<lower=1> N; // length of process5 array[N] real u; // 1/speed6 array[N] real v; // hoop distance7 matrix<lower=0>[K, K] alpha; // transit prior8 real<lower=0> tau; // sd u9 real<lower=0> rho; // sd v10}11parameters {12 simplex[K] theta1;13 simplex[K] theta2;14 // enforce an ordering: phi[1] <= phi[2]15 ordered[K] phi; // emission parameter for 1/speed16 ordered[K] lambda; // emission parameter for hoop dist17}18transformed parameters {19 array[K] simplex[K] theta; // transit probs20 theta[1] = theta1;21 theta[2] = theta2;22}23model {24 // priors25 for (k in 1 : K) {26 target += dirichlet_lpdf(theta[k] | alpha[k, : ]');27 }28 target += normal_lpdf(phi[1] | 0, 1);29 target += normal_lpdf(phi[2] | 3, 1);30 target += normal_lpdf(lambda[1] | 0, 1);31 target += normal_lpdf(lambda[2] | 3, 1);32 // forward algorithm33 {34 array[K] real acc;35 array[N, K] real gamma;36 for (k in 1 : K) {37 gamma[1, k] = normal_lpdf(u[1] | phi[k], tau)38 + normal_lpdf(v[1] | lambda[k], rho);39 }40 for (t in 2 : N) {41 for (k in 1 : K) {42 for (j in 1 : K) {43 acc[j] = gamma[t - 1, j] + log(theta[j, k])44 + normal_lpdf(u[t] | phi[k], tau)45 + normal_lpdf(v[t] | lambda[k], rho);46 }47 gamma[t, k] = log_sum_exp(acc);48 }49 }50 target += log_sum_exp(gamma[N]);51 }52}53generated quantities {54 array[N] int<lower=1, upper=K> z_star;55 real log_p_z_star;56 // Viterbi algorithm57 {58 array[N, K] int back_ptr;59 array[N, K] real best_logp;60 for (k in 1 : K) {61 best_logp[1, K] = normal_lpdf(u[1] | phi[k], tau)62 + normal_lpdf(v[1] | lambda[k], rho);63 }64 for (t in 2 : N) {65 for (k in 1 : K) {66 best_logp[t, k] = negative_infinity();67 for (j in 1 : K) {68 real logp;69 logp = best_logp[t - 1, j] + log(theta[j, k])70 + normal_lpdf(u[t] | phi[k], tau)71 + normal_lpdf(v[t] | lambda[k], rho);72 if (logp > best_logp[t, k]) {73 back_ptr[t, k] = j;74 best_logp[t, k] = logp;75 }76 }77 }78 }79 log_p_z_star = max(best_logp[N]);80 for (k in 1 : K) {81 if (best_logp[N, k] == log_p_z_star) {82 z_star[N] = k;83 }84 }85 for (t in 1 : (N - 1)) {86 z_star[N - t] = back_ptr[N - t + 1, z_star[N - t + 1]];87 }88 }89}9091//@ DATA { N: 416, K: 2, u: [416 values], v: [416 values], alpha: [2×2 matrix], tau: 0.1, rho: 0.1 } // values supplied at runtime92//@ PARAMS ["theta1[1]","theta1[2]","theta2[1]","theta2[2]","phi[1]","phi[2]","lambda[1]","lambda[2]"]93//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}94
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0008 ≤ tol 0.0026 · floors 0.0012/0.0013 |
Skeletal age is to be inferred for nChild = 13 children using nInd = 34 skeletal maturity indicators. For each indicator j (j = 1..34), the data provide: a discrimination parameter delta_j (a positive real), and up to 4 ordered threshold parameters gamma[j,1], ..., gamma[j, ncat_j - 1] (real-valued; -1 indicates missing). The number of ordered categories for indicator j is ncat_j (provided as data). Each child i's observed grade on indicator j is grade[i,j], an integer in {1, ..., ncat_j}, with -1 indicating a missing observation. The latent age theta_i for child i has a Normal(mean 0, sd 36) prior.
Each child i has an unknown latent skeletal age theta_i. For each non-missing observation of indicator j on child i, the cumulative probability of exceeding grade category k (for k = 1, ..., ncat_j - 1) is inv_logit(delta_j * (theta_i - gamma[j,k])). The probability of observing each grade category is derived from these cumulative probabilities: P(grade = 1) = 1 - Q_{j,1}, P(grade = k) = Q_{j,k-1} - Q_{j,k} for k = 2, ..., ncat_j - 1, and P(grade = ncat_j) = Q_{j, ncat_j - 1}, where Q_{j,k} = inv_logit(delta_j * (theta_i - gamma[j,k])). Missing grade observations contribute nothing to the likelihood.
The marginal posterior distribution of each parameter given the data: the latent skeletal ages theta_1, ..., theta_13 (one per child).
answer spec
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1/*2 * Note:3 * 1. Since it is just the response that is4 * modelled as categorical distribution,5 * we should be able to run the model now except handling6 * the missing. However, the data structure is a bit7 * difficult to deal with (Allowing some redundancy8 * in the transformed parameters (Q here), the model9 * is fine in Stan.10 * 2. The missing data is recoded as `-1`, which is11 * not modeled for `gamma` as in the OpenBUGS example12 * and not modeled for `grade`.13 */1415data {16 int<lower=0> nChild;17 int<lower=0> nInd;18 array[nInd, 4] real gamma; // -1 if missing19 array[nInd] real delta;20 array[nInd] int<lower=0> ncat;21 array[nChild, nInd] int grade; // -1 if missing22}23parameters {24 array[nChild] real theta;25}26model {27 array[nChild, nInd, 5] real p;28 array[nChild, nInd, 4] real Q;29 theta ~ normal(0.0, 36);30 for (i in 1 : nChild) {31 // Probability of observing grade k given theta32 for (j in 1 : nInd) {33 // Cumulative probability of > grade k given theta34 for (k in 1 : (ncat[j] - 1)) {35 Q[i, j, k] = inv_logit(delta[j] * (theta[i] - gamma[j, k]));36 }37 p[i, j, 1] = 1 - Q[i, j, 1];38 for (k in 2 : (ncat[j] - 1)) {39 p[i, j, k] = Q[i, j, k - 1] - Q[i, j, k];40 }41 p[i, j, ncat[j]] = Q[i, j, ncat[j] - 1];4243 // incement log probability directly because grade[i, j]44 // has categorical distribution with varying dimension.45 // for missing grade[i, j] = -1, there is no log prob46 // contribution47 if (grade[i, j] != -1) {48 target += log(p[i, j, grade[i, j]]);49 }50 }51 }52}5354//@ DATA { ncat: [34 values], nChild: 13, nInd: 34, grade: [13×34 matrix], delta: [34 values], gamma: [34×4 matrix] } // values supplied at runtime55//@ PARAMS ["theta[1]","theta[2]","theta[3]","theta[4]","theta[5]","theta[6]","theta[7]","theta[8]","theta[9]","theta[10]","theta[11]","theta[12]","theta[13]"]56//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}57
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| theta[1] | 0.338 ± 0.204 | — |
| theta[2] | 1.37 ± 0.255 | — |
| theta[3] | 2.35 ± 0.267 | — |
| theta[4] | 2.91 ± 0.291 | — |
| theta[5] | 5.57 ± 0.513 | — |
| theta[6] | 6.70 ± 0.584 | — |
| theta[7] | 6.49 ± 0.585 | — |
| theta[8] | 8.92 ± 0.708 | — |
| theta[9] | 8.95 ± 0.627 | — |
| theta[10] | 11.97 ± 0.661 | — |
| theta[11] | 11.53 ± 0.889 | — |
| theta[12] | 15.80 ± 0.557 | — |
| theta[13] | 17.00 ± 0.741 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0405 ≤ tol 0.2048 · floors 0.0400/0.0794 |
For N observations of diamonds, the data provide a response variable Y containing log-transformed prices and a design matrix X with N rows and K = 25 columns. The first column of X is all 1s (for the intercept). The remaining 24 columns contain numerical and categorical predictor variables. The model centers these 24 predictors by subtracting each column's mean. A binary flag prior_only indicates whether the likelihood should be ignored (prior-only inference). The 24 regression coefficients b[1] through b[24] have independent standard Normal(0, 1) priors. The intercept parameter has a Student-t prior with 3 degrees of freedom, location 8, and scale 10. The residual standard deviation sigma, constrained positive, has a half-Student-t prior with 3 degrees of freedom, location 0, and scale 10 (a Student-t(3, 0, 10) prior truncated to the positive reals).
Each observation's log price is normally distributed with a mean equal to the intercept plus a linear combination of the 24 centered predictors weighted by their respective coefficients, and a common standard deviation sigma across all observations. The centering of predictors is performed by subtracting each predictor's column mean from all observations before forming the linear predictor. This results in the intercept representing the expected log price when all centered predictors are at zero, which corresponds to the predictors being at their observed column means.
The marginal posterior distributions of the 26 parameters: the 24 centered predictor coefficients b[1], b[2], ..., b[24], the intercept (reported as Intercept), and the residual standard deviation sigma.
answer spec
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"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// generated with brms 2.10.023functions {45}6data {7 int<lower=1> N; // number of observations8 vector[N] Y; // response variable9 int<lower=1> K; // number of population-level effects10 matrix[N, K] X; // population-level design matrix11 int prior_only; // should the likelihood be ignored?12}13transformed data {14 int Kc = K - 1;15 matrix[N, Kc] Xc; // centered version of X without an intercept16 vector[Kc] means_X; // column means of X before centering17 for (i in 2 : K) {18 means_X[i - 1] = mean(X[ : , i]);19 Xc[ : , i - 1] = X[ : , i] - means_X[i - 1];20 }21}22parameters {23 vector[Kc] b; // population-level effects24 // temporary intercept for centered predictors25 real Intercept;26 real<lower=0> sigma; // residual SD27}28transformed parameters {2930}31model {32 // priors including all constants33 target += normal_lpdf(b | 0, 1);34 target += student_t_lpdf(Intercept | 3, 8, 10);35 target += student_t_lpdf(sigma | 3, 0, 10)36 - 1 * student_t_lccdf(0 | 3, 0, 10);37 // likelihood including all constants38 if (!prior_only) {39 target += normal_id_glm_lpdf(Y | Xc, Intercept, b, sigma);40 }41}42generated quantities {43 // actual population-level intercept44 real b_Intercept = Intercept - dot_product(means_X, b);45}4647//@ DATA { N: 5000, Y: [5000 values], K: 25, X: [5000×25 matrix], prior_only: 0 } // values supplied at runtime48//@ PARAMS ["b[1]","b[2]","b[3]","b[4]","b[5]","b[6]","b[7]","b[8]","b[9]","b[10]","b[11]","b[12]","b[13]","b[14]","b[15]","b[16]","b[17]","b[18]","b[19]","b[20]","b[21]","b[22]","b[23]","b[24]","Intercept","sigma"]49//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}50
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| b[1] | 6.64 ± 0.271 | 6.66 ± 0.235 |
| b[2] | 6.35 ± 0.322 | 6.38 ± 0.304 |
| b[3] | -4.67 ± 0.314 | -4.70 ± 0.302 |
| b[4] | 1.46 ± 0.149 | 1.44 ± 0.136 |
| b[5] | 0.134 ± 0.008 | 0.135 ± 0.008 |
| b[6] | -0.040 ± 0.007 | -0.041 ± 0.007 |
| b[7] | 0.023 ± 0.006 | 0.023 ± 0.006 |
| b[8] | 0.002 ± 0.004 | 0.002 ± 0.004 |
| b[9] | -0.444 ± 0.006 | -0.445 ± 0.006 |
| b[10] | -0.093 ± 0.006 | -0.093 ± 0.005 |
| b[11] | -0.012 ± 0.005 | -0.013 ± 0.005 |
| b[12] | 0.011 ± 0.005 | 0.011 ± 0.005 |
| b[13] | -0.002 ± 0.005 | -0.002 ± 0.004 |
| b[14] | 6.02e-4 ± 0.004 | 9.84e-4 ± 0.004 |
| b[15] | 0.900 ± 0.011 | 0.901 ± 0.011 |
| b[16] | -0.221 ± 0.010 | -0.221 ± 0.010 |
| b[17] | 0.131 ± 0.009 | 0.131 ± 0.009 |
| b[18] | -0.058 ± 0.007 | -0.057 ± 0.007 |
| b[19] | 0.018 ± 0.005 | 0.018 ± 0.006 |
| b[20] | -0.002 ± 0.005 | -0.002 ± 0.005 |
| b[21] | 0.032 ± 0.004 | 0.032 ± 0.004 |
| b[22] | -6.09 ± 0.293 | -6.12 ± 0.291 |
| b[23] | 4.62 ± 0.290 | 4.63 ± 0.289 |
| b[24] | -1.45 ± 0.149 | -1.43 ± 0.154 |
| Intercept | 7.79 ± 0.002 | 7.79 ± 0.002 |
| sigma | 0.123 ± 0.001 | 0.123 ± 0.001 |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0141 ≤ tol 0.0478 · floors 0.0239/0.0236 |
A shuttle-avoidance experiment was run on 30 dogs over 25 trials each. The binary outcome for dog i on trial t is provided as data, equal to 1 if the dog avoided the shock on that trial and 0 otherwise. Three coefficients beta_1, beta_2, beta_3 each have a Normal(mean 0, sd 100) prior, independently.
For each dog and trial, let a be the number of prior trials on which that dog avoided the shock and s be the number of prior trials on which it was shocked (both counted over the earlier trials for that dog, starting at zero on the first trial). The log-odds that the dog avoids on the current trial is beta_1 + beta_2 * a + beta_3 * s. The outcome is drawn from a Bernoulli distribution with that success probability on the log-odds scale.
The marginal posterior distribution of each coefficient given the data: beta_1, beta_2, and beta_3.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> n_dogs;3 int<lower=0> n_trials;4 array[n_dogs, n_trials] int<lower=0, upper=1> y;5}6parameters {7 vector[3] beta;8}9transformed parameters {10 matrix[n_dogs, n_trials] n_avoid;11 matrix[n_dogs, n_trials] n_shock;12 matrix[n_dogs, n_trials] p;1314 for (j in 1 : n_dogs) {15 n_avoid[j, 1] = 0;16 n_shock[j, 1] = 0;17 for (t in 2 : n_trials) {18 n_avoid[j, t] = n_avoid[j, t - 1] + 1 - y[j, t - 1];19 n_shock[j, t] = n_shock[j, t - 1] + y[j, t - 1];20 }21 for (t in 1 : n_trials) {22 p[j, t] = beta[1] + beta[2] * n_avoid[j, t] + beta[3] * n_shock[j, t];23 }24 }25}26model {27 beta ~ normal(0, 100);28 for (i in 1 : n_dogs) {29 for (j in 1 : n_trials) {30 y[i, j] ~ bernoulli_logit(p[i, j]);31 }32 }33}3435//@ DATA { n_dogs: 30, n_trials: 25, y: [30×25 matrix] } // values supplied at runtime36//@ PARAMS ["beta[1]","beta[2]","beta[3]"]37//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}38
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0073 ≤ tol 0.0429 · floors 0.0139/0.0135 |
A shuttle-avoidance experiment was run on 30 dogs over 25 trials each. The binary outcome for dog j on trial t is provided as data, equal to 1 if the dog avoided the shock on that trial and 0 otherwise. The avoidance-decay parameter a lies in (0, 1) with a Uniform(0, 1) prior. The shock-sensitivity parameter b lies in (0, 1) with a Uniform(0, 1) prior.
For each dog j and trial t, let s be the number of prior trials on which that dog was shocked and v the number on which it avoided (both counted over the earlier trials for that dog, starting at zero on the first trial). The probability that the dog avoids on the current trial is a raised to the power s, times b raised to the power v. The outcome is drawn from a Bernoulli distribution with that probability.
The marginal posterior distribution of each parameter given the data: the avoidance-decay parameter a and the shock-sensitivity parameter b.
answer spec
{
"kind": "record",
"fields": {
"a": {
"kind": "dist",
"domain": "real"
},
"b": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> n_dogs;3 int<lower=0> n_trials;4 array[n_dogs, n_trials] int<lower=0, upper=1> y;5}6transformed data {7 int<lower=0> J = n_dogs;8 int<lower=0> T = n_trials;9 matrix<lower=0>[J, T] prev_shock;10 matrix<lower=0>[J, T] prev_avoid;1112 for (j in 1 : J) {13 prev_shock[j, 1] = 0;14 prev_avoid[j, 1] = 0;15 for (t in 2 : T) {16 prev_shock[j, t] = prev_shock[j, t - 1] + y[j, t - 1];17 prev_avoid[j, t] = prev_avoid[j, t - 1] + 1 - y[j, t - 1];18 }19 }20}21parameters {22 real<lower=0, upper=1> a;23 real<lower=0, upper=1> b;24}25model {26 for (j in 1 : J) {27 for (t in 1 : T) {28 real p = a ^ prev_shock[j, t] * b ^ prev_avoid[j, t];29 y[j, t] ~ bernoulli(p);30 }31 }32}33generated quantities {34 array[n_dogs, n_trials] int<lower=0, upper=1> y_rep;35 {36 real prev_shock_rep;37 real prev_avoid_rep;38 real p_rep;39 for (j in 1 : J) {40 prev_shock_rep = 0;41 prev_avoid_rep = 0;42 y_rep[j, 1] = 1;43 for (t in 2 : T) {44 prev_shock_rep = prev_shock_rep + y_rep[j, t - 1];45 prev_avoid_rep = prev_avoid_rep + 1 - y_rep[j, t - 1];46 p_rep = a ^ prev_shock_rep * b ^ prev_avoid_rep;47 y_rep[j, t] = bernoulli_rng(p_rep);48 }49 }50 }51}5253//@ DATA { n_dogs: 30, n_trials: 25, y: [30×25 matrix] } // values supplied at runtime54//@ PARAMS ["a","b"]55//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}56
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0007 ≤ tol 0.0039 · floors 0.0013/0.0010 |
A shuttle-avoidance experiment was run on 30 dogs over 25 trials each. The binary outcome for dog i on trial t is provided as data, equal to 1 if the dog avoided the shock on that trial and 0 otherwise. The coefficient beta_1 has a Uniform(-100, 0) prior and the coefficient beta_2 has a Uniform(0, 100) prior.
For each dog and trial, let a be the number of prior trials on which that dog avoided the shock and s be the number of prior trials on which it was shocked (both counted over the earlier trials for that dog, starting at zero on the first trial). The probability that the dog avoids on the current trial is the inverse-logit of beta_1 * a + beta_2 * s. The outcome is drawn from a Bernoulli distribution with that probability.
The marginal posterior distribution of each coefficient given the data: beta_1 and beta_2.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> n_trials;3 int<lower=0> n_dogs;4 array[n_dogs, n_trials] int<lower=0, upper=1> y;5}6parameters {7 vector[2] beta;8}9transformed parameters {10 matrix[n_dogs, n_trials] n_avoid;11 matrix[n_dogs, n_trials] n_shock;12 matrix[n_dogs, n_trials] p;1314 for (j in 1 : n_dogs) {15 n_avoid[j, 1] = 0;16 n_shock[j, 1] = 0;17 for (t in 2 : n_trials) {18 n_avoid[j, t] = n_avoid[j, t - 1] + 1 - y[j, t - 1];19 n_shock[j, t] = n_shock[j, t - 1] + y[j, t - 1];20 }21 for (t in 1 : n_trials) {22 p[j, t] = inv_logit(beta[1] * n_avoid[j, t] + beta[2] * n_shock[j, t]);23 }24 }25}26model {27 beta[1] ~ uniform(-100, 0);28 beta[2] ~ uniform(0, 100);29 for (i in 1 : n_dogs) {30 for (j in 1 : n_trials) {31 y[i, j] ~ bernoulli(p[i, j]);32 }33 }34}3536//@ DATA { n_dogs: 30, n_trials: 25, y: [30×25 matrix] } // values supplied at runtime37//@ PARAMS ["beta[1]","beta[2]"]38//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}39
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0017 ≤ tol 0.0070 · floors 0.0026/0.0029 |
Length and age measurements are available for N = 27 dugongs: an array of ages x_i and an array of body lengths Y_i (i = 1..27). The intercept parameter alpha has a Normal(mean 0, sd 1000) prior. The growth-amplitude parameter beta has a Normal(mean 0, sd 1000) prior. The decay-rate parameter lambda is constrained to lie in [0.5, 1] and has a Uniform(0.5, 1) prior. The precision parameter tau is constrained to be positive and has a Gamma(shape 0.0001, rate 0.0001) prior.
The expected length of dugong i is alpha minus beta times lambda raised to the power x_i, giving a nonlinear growth curve that approaches a limiting length asymptotically as age increases. The observed length Y_i is drawn from a Normal distribution centered on that expected value with standard deviation sigma = 1 / sqrt(tau). Two derived quantities are also of interest: sigma = 1 / sqrt(tau) and U3 = logit(lambda).
The marginal posterior distribution of each parameter and derived quantity given the data: the intercept alpha, the growth amplitude beta, the decay rate lambda, the precision tau, the standard deviation sigma, and the logit-transformed decay rate U3.
answer spec
{
"kind": "record",
"fields": {
"alpha": {
"kind": "dist",
"domain": "real"
},
"beta": {
"kind": "dist",
"domain": "real"
},
"lambda": {
"kind": "dist",
"domain": "real"
},
"tau": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
},
"U3": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 array[N] real x;4 array[N] real Y;5}6parameters {7 real alpha;8 real beta;9 real<lower=.5, upper=1> lambda; // orginal gamma in the JAGS example10 real<lower=0> tau;11}12transformed parameters {13 real sigma;14 real U3;15 sigma = 1 / sqrt(tau);16 U3 = logit(lambda);17}18model {19 array[N] real m;20 for (i in 1 : N) {21 m[i] = alpha - beta * pow(lambda, x[i]);22 }23 Y ~ normal(m, sigma);2425 alpha ~ normal(0.0, 1000);26 beta ~ normal(0.0, 1000);27 lambda ~ uniform(.5, 1);28 tau ~ gamma(.0001, .0001);29}3031//@ DATA { Y: [27 values], x: [27 values], N: 27 } // values supplied at runtime32//@ PARAMS ["alpha","beta","lambda","tau","sigma","U3"]33//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}34
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.9714 ≤ tol 3.3682 · floors 1.6768/1.2891 |
For each of N individuals, the data provide the person's earnings and height in inches. The regression has two coefficients, an intercept and a slope on height, each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each individual's earnings are normally distributed with a mean equal to the intercept plus the slope times that individual's height, and a common standard deviation sigma across all individuals.
The marginal posterior distribution of each of the three parameters: the intercept (reported as beta[1]), the slope on height (reported as beta[2]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] earn;4 vector[N] height;5}6parameters {7 vector[2] beta;8 real<lower=0> sigma;9}10model {11 earn ~ normal(beta[1] + beta[2] * height, sigma);12}1314//@ DATA { N: 1192, earn: [1192 values], height: [1192 values] } // values supplied at runtime15//@ PARAMS ["beta[1]","beta[2]","sigma"]16//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}17
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=364.2235 ≤ tol 1066.3760 · floors 533.1880/439.1448 |
For each of N individuals, the data provide that individual's earnings (a positive real value in dollars) and height (a real-valued measurement in inches). The model operates on a log base 10 transformation of earnings. The regression has two coefficients, an intercept and a slope on height, each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
For each individual, we compute the log base 10 of that individual's earnings. The log base 10 transformed earnings are normally distributed with mean equal to the intercept beta[1] plus the slope beta[2] times that individual's height, and with a common standard deviation sigma across all individuals.
The marginal posterior distribution of each of the three parameters: beta[1] (the intercept), beta[2] (the height coefficient), and sigma (the error standard deviation).
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] earn;4 vector[N] height;5}6transformed data {7 // log 10 transformation8 vector[N] log10_earn;9 for (i in 1 : N) {10 log10_earn[i] = log10(earn[i]);11 }12}13parameters {14 vector[2] beta;15 real<lower=0> sigma;16}17model {18 log10_earn ~ normal(beta[1] + beta[2] * height, sigma);19}2021//@ DATA { N: 1192, earn: [1192 values], height: [1192 values] } // values supplied at runtime22//@ PARAMS ["beta[1]","beta[2]","sigma"]23//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}24
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0072 ≤ tol 0.0320 · floors 0.0087/0.0092 |
For each of N = 1192 adults the data provide annual earnings (in dollars, positive) and height (in inches). The model is fit to the natural logarithm of earnings. The regression has two coefficients, an intercept and a slope on height, each with a flat (improper uniform) prior over the real line; the error standard deviation sigma, constrained positive, also has a flat (improper uniform) prior.
The natural logarithm of each person's earnings is Normal-distributed with a mean equal to the intercept plus the slope times that person's height, and a common standard deviation sigma across all people.
The marginal posterior distribution of each of the three parameters, all on the log-earnings scale: the intercept (reported as beta[1]), the slope on height (reported as beta[2]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] earn;4 vector[N] height;5}6transformed data {7 // log transformation8 vector[N] log_earn;9 log_earn = log(earn);10}11parameters {12 vector[2] beta;13 real<lower=0> sigma;14}15model {16 log_earn ~ normal(beta[1] + beta[2] * height, sigma);17}1819//@ DATA { N: 1192, earn: [1192 values], height: [1192 values] } // values supplied at runtime20//@ PARAMS ["beta[1]","beta[2]","sigma"]21//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}22
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0168 ≤ tol 0.0765 · floors 0.0382/0.0189 |
For each of N = 1192 individuals the data provide earnings in dollars and two predictors: height measured in inches and a binary indicator of male gender. The regression has three coefficients—an intercept, a slope on height, and a slope on the male indicator—each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
The natural logarithm of each individual's earnings is normally distributed with a mean equal to the intercept plus the height coefficient times that individual's height plus the male coefficient times that individual's gender indicator, and a common standard deviation sigma across all individuals.
The marginal posterior distribution of each of the four parameters: the intercept (reported as beta[1]), the slope on height (reported as beta[2]), the slope on the male indicator (reported as beta[3]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] earn;4 vector[N] height;5 vector[N] male;6}7transformed data {8 // log transformation9 vector[N] log_earn;10 log_earn = log(earn);11}12parameters {13 vector[3] beta;14 real<lower=0> sigma;15}16model {17 log_earn ~ normal(beta[1] + beta[2] * height + beta[3] * male, sigma);18}1920//@ DATA { N: 1192, earn: [1192 values], height: [1192 values], male: [1192 values] } // values supplied at runtime21//@ PARAMS ["beta[1]","beta[2]","beta[3]","sigma"]22//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}23
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0340 ≤ tol 0.0895 · floors 0.0448/0.0294 |
For each of N individuals, the data provide the individual's earnings (a positive real-valued number), height in inches, and a binary indicator of male gender (1 = male, 0 = female). The model operates on the natural logarithm of earnings. The regression has four coefficients—an intercept, a slope on height, a slope on the male indicator, and a slope on the height-male interaction—each with a flat (improper uniform) prior over the real line. The residual standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each individual's log earnings is normally distributed with a mean equal to the intercept plus the height coefficient times that individual's height, plus the male coefficient times that individual's male indicator, plus the interaction coefficient times the product of height and male indicator. The standard deviation of this normal distribution is sigma, common across all individuals.
The marginal posterior distributions of the five parameters: the intercept (beta[1]), the height coefficient (beta[2]), the male indicator coefficient (beta[3]), the height-male interaction coefficient (beta[4]), and the residual standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] earn;4 vector[N] height;5 vector[N] male;6}7transformed data {8 vector[N] log_earn; // log transformation9 vector[N] inter; // interaction10 log_earn = log(earn);11 inter = height .* male;12}13parameters {14 vector[4] beta;15 real<lower=0> sigma;16}17model {18 log_earn ~ normal(beta[1] + beta[2] * height + beta[3] * male19 + beta[4] * inter, sigma);20}2122//@ DATA { N: 1192, earn: [1192 values], height: [1192 values], male: [1192 values] } // values supplied at runtime23//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","sigma"]24//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}25
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0429 ≤ tol 0.2225 · floors 0.0904/0.0758 |
For each of N = 1192 individuals, the data provide raw earnings (positive reals), height in inches, and a binary male indicator (1 for male, 0 for female). The model operates on log-transformed earnings and standardized height (computed by subtracting the sample mean and dividing by the sample standard deviation of the height data). An interaction term is formed as the product of standardized height and the male indicator. All four regression coefficients beta[1], beta[2], beta[3], beta[4] have a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals (0, infinity).
For each individual, the natural logarithm of earnings is normally distributed with a mean equal to the sum of an intercept (beta[1]), a coefficient (beta[2]) times the person's standardized height, a coefficient (beta[3]) times the male indicator, and a coefficient (beta[4]) times the interaction between standardized height and male indicator, with a common standard deviation sigma across all individuals.
The marginal posterior distributions of the five parameters: beta[1] (the intercept), beta[2] (the standardized height coefficient), beta[3] (the male indicator coefficient), beta[4] (the interaction coefficient), and sigma (the error standard deviation).
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] earn;4 vector[N] height;5 vector[N] male;6}7transformed data {8 vector[N] log_earn; // log transformation9 vector[N] z_height; // standardization10 vector[N] inter; // interaction11 log_earn = log(earn);12 z_height = (height - mean(height)) / sd(height);13 inter = z_height .* male;14}15parameters {16 vector[4] beta;17 real<lower=0> sigma;18}19model {20 log_earn ~ normal(beta[1] + beta[2] * z_height + beta[3] * male21 + beta[4] * inter, sigma);22}2324//@ DATA { N: 1192, earn: [1192 values], height: [1192 values], male: [1192 values] } // values supplied at runtime25//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","sigma"]26//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}27
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0025 ≤ tol 0.0119 · floors 0.0043/0.0039 |
For each of N = 1192 workers the data provide the worker's earnings (a positive real value), height in inches (a positive real value), and a binary indicator of male gender. The model operates on the natural logarithm of earnings and the natural logarithm of height. The regression has three coefficients: an intercept, a slope on log-transformed height, and a slope on the male indicator, each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each worker's log-transformed earnings is Normal-distributed with a mean equal to the intercept plus the slope for log-transformed height times that worker's log-transformed height, plus the slope for male status times that worker's male indicator (0 if female, 1 if male), and a common standard deviation sigma across all workers.
The marginal posterior distribution of each of the four parameters: the intercept (reported as beta[1]), the slope on log-transformed height (reported as beta[2]), the slope on the male indicator (reported as beta[3]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] earn;4 vector[N] height;5 vector[N] male;6}7transformed data {8 vector[N] log_earn; // log transformations9 vector[N] log_height;10 log_earn = log(earn);11 log_height = log(height);12}13parameters {14 vector[3] beta;15 real<lower=0> sigma;16}17model {18 // vectorization19 log_earn ~ normal(beta[1] + beta[2] * log_height + beta[3] * male, sigma);20}2122//@ DATA { N: 1192, earn: [1192 values], height: [1192 values], male: [1192 values] } // values supplied at runtime23//@ PARAMS ["beta[1]","beta[2]","beta[3]","sigma"]24//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}25
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0910 ≤ tol 0.4362 · floors 0.2181/0.1353 |
Eight schools each ran an SAT-coaching program. For school j (j = 1..8) a separate analysis produced an estimated treatment effect y_j on test scores together with the known standard error sigma_j of that estimate (each sigma_j is positive); the eight estimates and their eight standard errors are provided as data (J = 8). The population mean mu has a Normal(mean 0, sd 5) prior. The population standard deviation tau, constrained to be positive, has a half-Cauchy(location 0, scale 5) prior.
Each school has an unknown true coaching effect theta_j, drawn independently from a Normal distribution with mean mu and standard deviation tau. The observed estimate y_j is then Normal-distributed around that true effect theta_j with the school's known standard error sigma_j.
The marginal posterior distribution of each parameter given the data: the population mean mu, the population standard deviation tau, and the eight true school effects theta_1, ..., theta_8.
answer spec
{
"kind": "record",
"fields": {
"theta[1]": {
"kind": "dist",
"domain": "real"
},
"theta[2]": {
"kind": "dist",
"domain": "real"
},
"theta[3]": {
"kind": "dist",
"domain": "real"
},
"theta[4]": {
"kind": "dist",
"domain": "real"
},
"theta[5]": {
"kind": "dist",
"domain": "real"
},
"theta[6]": {
"kind": "dist",
"domain": "real"
},
"theta[7]": {
"kind": "dist",
"domain": "real"
},
"theta[8]": {
"kind": "dist",
"domain": "real"
},
"mu": {
"kind": "dist",
"domain": "real"
},
"tau": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> J; // number of schools3 array[J] real y; // estimated treatment4 array[J] real<lower=0> sigma; // std of estimated effect5}6parameters {7 vector[J] theta_trans; // transformation of theta8 real mu; // hyper-parameter of mean9 real<lower=0> tau; // hyper-parameter of sd10}11transformed parameters {12 vector[J] theta;13 // original theta14 theta = theta_trans * tau + mu;15}16model {17 theta_trans ~ normal(0, 1);18 y ~ normal(theta, sigma);19 mu ~ normal(0, 5); // a non-informative prior20 tau ~ cauchy(0, 5);21}2223//@ DATA { J: 8, y: [8 values], sigma: [8 values] } // values supplied at runtime24//@ PARAMS ["theta[1]","theta[2]","theta[3]","theta[4]","theta[5]","theta[6]","theta[7]","theta[8]","mu","tau"]25//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}26
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.2568 ≤ tol 0.8077 · floors 0.3903/0.4039 |
The data consist of T time series observations y[1], y[2], ..., y[T] of real values, and a positive real initial conditional standard deviation sigma1 at time 1. The model has four parameters, each with a flat improper uniform prior: mu over the real line; alpha0 constrained positive over (0, infinity) with a flat improper uniform prior; alpha1 constrained to the interval [0, 1] with a flat improper uniform prior; and beta1 constrained to [0, 1 - alpha1] with a flat improper uniform prior.
This is a GARCH(1,1) model for conditional heteroscedasticity in time series. The conditional standard deviation at time 1 is fixed at the given value sigma1. For each time t from 2 to T, the conditional standard deviation evolves according to the recursion: sigma[t] equals the square root of alpha0 plus alpha1 times the squared deviation of the previous observation from mu, plus beta1 times the square of the previous conditional standard deviation. Each observation y[t] is generated from a normal distribution with mean mu and standard deviation sigma[t].
The marginal posterior distributions of the four parameters: mu (the mean), alpha0 (the intercept of the conditional variance), alpha1 (the lagged squared residual coefficient), and beta1 (the lagged conditional variance coefficient).
answer spec
{
"kind": "record",
"fields": {
"mu": {
"kind": "dist",
"domain": "real"
},
"alpha0": {
"kind": "dist",
"domain": "real"
},
"alpha1": {
"kind": "dist",
"domain": "real"
},
"beta1": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> T;3 array[T] real y;4 real<lower=0> sigma1;5}6parameters {7 real mu;8 real<lower=0> alpha0;9 real<lower=0, upper=1> alpha1;10 real<lower=0, upper=(1 - alpha1)> beta1;11}12model {13 array[T] real sigma;14 sigma[1] = sigma1;15 for (t in 2 : T) {16 sigma[t] = sqrt(alpha0 + alpha1 * square(y[t - 1] - mu)17 + beta1 * square(sigma[t - 1]));18 }1920 y ~ normal(mu, sigma);21}2223//@ DATA { T: 200, y: [200 values], sigma1: 0.5 } // values supplied at runtime24//@ PARAMS ["mu","alpha0","alpha1","beta1"]25//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}26
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0185 ≤ tol 0.0708 · floors 0.0197/0.0302 |
For N = 11 observations, the data provide an input location x_i (a real number) and a count observation k_i (a non-negative integer) for each observation i. The model has two hyperparameters: rho, a length scale parameter constrained to be positive, with a gamma(25, 4) prior; and alpha, a marginal standard deviation parameter constrained to be positive, with a half-normal prior truncated to the positive reals (equivalently, a normal(0, 2) prior restricted to alpha > 0).
The observed counts follow a Poisson regression with latent Gaussian process. For each observation i, the count k_i is Poisson-distributed with log-rate parameter f_i, where f is a latent one-dimensional Gaussian process evaluated at the input locations x_1, ..., x_N. The Gaussian process is specified by an exponential-quadratic (squared exponential) covariance kernel with length-scale parameter rho and marginal standard deviation parameter alpha. The latent GP values f_1, ..., f_N are generated from a multivariate normal distribution with mean zero and covariance matrix determined by evaluating the kernel at the observed input locations.
The marginal posterior distributions of the two hyperparameters rho (length scale) and alpha (marginal standard deviation), and the latent Gaussian process values f[1], f[2], f[3], f[4], f[5], f[6], f[7], f[8], f[9], f[10], f[11].
answer spec
{
"kind": "record",
"fields": {
"rho": {
"kind": "dist",
"domain": "real"
},
"alpha": {
"kind": "dist",
"domain": "real"
},
"f[1]": {
"kind": "dist",
"domain": "real"
},
"f[2]": {
"kind": "dist",
"domain": "real"
},
"f[3]": {
"kind": "dist",
"domain": "real"
},
"f[4]": {
"kind": "dist",
"domain": "real"
},
"f[5]": {
"kind": "dist",
"domain": "real"
},
"f[6]": {
"kind": "dist",
"domain": "real"
},
"f[7]": {
"kind": "dist",
"domain": "real"
},
"f[8]": {
"kind": "dist",
"domain": "real"
},
"f[9]": {
"kind": "dist",
"domain": "real"
},
"f[10]": {
"kind": "dist",
"domain": "real"
},
"f[11]": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=1> N;3 array[N] real x;4 array[N] int k;5}6parameters {7 real<lower=0> rho;8 real<lower=0> alpha;9 vector[N] f_tilde;10}11transformed parameters {12 vector[N] f;13 {14 matrix[N, N] cov = gp_exp_quad_cov(x, alpha, rho)15 + diag_matrix(rep_vector(1e-10, N));16 matrix[N, N] L_cov = cholesky_decompose(cov);17 f = L_cov * f_tilde;18 }19}20model {21 rho ~ gamma(25, 4);22 alpha ~ normal(0, 2);23 f_tilde ~ normal(0, 1);2425 k ~ poisson_log(f);26}2728//@ DATA { N: 11, x: [11 values], k: [11 values] } // values supplied at runtime29//@ PARAMS ["rho","alpha","f[1]","f[2]","f[3]","f[4]","f[5]","f[6]","f[7]","f[8]","f[9]","f[10]","f[11]"]30//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}31
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| rho | 5.68 ± 0.694 | 5.61 ± 0.667 |
| alpha | 3.00 ± 0.816 | 2.89 ± 0.736 |
| f[1] | 3.64 ± 0.156 | 3.64 ± 0.154 |
| f[2] | 3.68 ± 0.128 | 3.69 ± 0.128 |
| f[3] | 3.25 ± 0.146 | 3.26 ± 0.143 |
| f[4] | 2.41 ± 0.202 | 2.42 ± 0.189 |
| f[5] | 1.57 ± 0.246 | 1.58 ± 0.240 |
| f[6] | 1.30 ± 0.260 | 1.31 ± 0.268 |
| f[7] | 1.94 ± 0.222 | 1.93 ± 0.236 |
| f[8] | 3.18 ± 0.146 | 3.17 ± 0.155 |
| f[9] | 4.25 ± 0.097 | 4.25 ± 0.098 |
| f[10] | 4.42 ± 0.086 | 4.41 ± 0.089 |
| f[11] | 3.51 ± 0.164 | 3.51 ± 0.164 |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0370 ≤ tol 0.1404 · floors 0.0666/0.0702 |
The data comprise N = 11 paired observations of input locations x and output values y. Each input x is a real number, and each output y is a real-valued observation. The model is fit to estimate three parameters: rho, a length-scale parameter constrained positive with a gamma(shape 25, rate 4) prior; alpha, the marginal standard deviation of the Gaussian process, constrained positive with a normal(mean 0, standard deviation 2) prior; and sigma, the noise standard deviation, constrained positive with a normal(mean 0, standard deviation 1) prior.
The observed outputs y follow a multivariate normal distribution with mean vector 0 and covariance matrix formed as K plus sigma times the identity matrix, where K is the N by N covariance matrix computed using an exponential quadratic (squared exponential) kernel evaluated at the input locations x, with hyperparameters alpha (the amplitude or marginal standard deviation) and rho (the length scale). The observations are thus jointly normal with zero mean and covariance K + sigma*I, where the diagonal elements of the noise contribution are sigma (not sigma squared), corresponding to a noise standard deviation of sigma on each observation.
The marginal posterior distributions of the three parameters: rho (the length scale), alpha (the Gaussian process amplitude), and sigma (the noise standard deviation).
answer spec
{
"kind": "record",
"fields": {
"rho": {
"kind": "dist",
"domain": "real"
},
"alpha": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=1> N;3 array[N] real x;4 vector[N] y;5}6parameters {7 real<lower=0> rho;8 real<lower=0> alpha;9 real<lower=0> sigma;10}11model {12 matrix[N, N] cov = gp_exp_quad_cov(x, alpha, rho)13 + diag_matrix(rep_vector(sigma, N));14 matrix[N, N] L_cov = cholesky_decompose(cov);1516 rho ~ gamma(25, 4);17 alpha ~ normal(0, 2);18 sigma ~ normal(0, 1);1920 y ~ multi_normal_cholesky(rep_vector(0, N), L_cov);21}2223//@ DATA { N: 11, x: [11 values], y: [11 values] } // values supplied at runtime24//@ PARAMS ["rho","alpha","sigma"]25//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}26
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0563 ≤ tol 0.2085 · floors 0.0645/0.0650 |
For a sequence of N observations indexed t = 1 to N, with K = 2 hidden states, the data provide an array y of N real-valued observations. The model has two transition probability vectors: theta1, the row of transition probabilities from state 1 (a probability vector summing to 1 with each component non-negative); and theta2, the row of transition probabilities from state 2 (similarly, a probability vector summing to 1). Both theta1 and theta2 have flat (improper uniform) priors. The model also has state-specific emission means mu[1] and mu[2], constrained so that mu[1] <= mu[2] and both positive. These means have Normal priors: mu[1] ~ Normal(3, 1) and mu[2] ~ Normal(10, 1).
A Hidden Markov Model generates the sequence of observations. At time t = 1, the hidden state is drawn uniformly at random from the K states (no explicit prior). At each subsequent time t = 2, 3, ..., N, the hidden state transitions according to the transition probabilities: if the state at time t-1 is state j, the state at time t is drawn from the categorical distribution with probabilities given by theta[j] (either theta1 if j=1, or theta2 if j=2). Given the hidden state k at time t, the observation y[t] is drawn from a Normal distribution with mean mu[k] and fixed standard deviation 1. Thus each observation y[t] ~ Normal(mu[state[t]], 1), where state[t] evolves according to the Markov transition probabilities across the entire sequence.
The marginal posterior distributions of the five parameters: theta1[1] and theta1[2] (the transition probabilities from state 1), theta2[1] and theta2[2] (the transition probabilities from state 2), and mu[1] and mu[2] (the state-specific observation means, ordered so that mu[1] <= mu[2]).
answer spec
{
"kind": "record",
"fields": {
"theta1[1]": {
"kind": "dist",
"domain": "real"
},
"theta1[2]": {
"kind": "dist",
"domain": "real"
},
"theta2[1]": {
"kind": "dist",
"domain": "real"
},
"theta2[2]": {
"kind": "dist",
"domain": "real"
},
"mu[1]": {
"kind": "dist",
"domain": "real"
},
"mu[2]": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// simple hmm example (1 output; 2 states)2data {3 int<lower=0> N;4 int<lower=0> K;5 array[N] real y;6}7parameters {8 simplex[K] theta1;9 simplex[K] theta2;10 // real mu[K];11 positive_ordered[K] mu;12}13transformed parameters {14 array[K] simplex[K] theta;15 theta[1] = theta1;16 theta[2] = theta2;17}18model {19 // priors20 target += normal_lpdf(mu[1] | 3, 1);21 target += normal_lpdf(mu[2] | 10, 1);22 // forward algorithm23 {24 array[K] real acc;25 array[N, K] real gamma;26 for (k in 1 : K) {27 gamma[1, k] = normal_lpdf(y[1] | mu[k], 1);28 }29 for (t in 2 : N) {30 for (k in 1 : K) {31 for (j in 1 : K) {32 acc[j] = gamma[t - 1, j] + log(theta[j, k])33 + normal_lpdf(y[t] | mu[k], 1);34 }35 gamma[t, k] = log_sum_exp(acc);36 }37 }38 target += log_sum_exp(gamma[N]);39 }40}41generated quantities {42 array[N] int<lower=1, upper=K> z_star;43 real log_p_z_star;44 {45 array[N, K] int back_ptr;46 array[N, K] real best_logp;47 for (k in 1 : K) {48 best_logp[1, k] = normal_lpdf(y[1] | mu[k], 1);49 }50 for (t in 2 : N) {51 for (k in 1 : K) {52 best_logp[t, k] = negative_infinity();53 for (j in 1 : K) {54 real logp;55 logp = best_logp[t - 1, j] + log(theta[j, k])56 + normal_lpdf(y[t] | mu[k], 1);57 if (logp > best_logp[t, k]) {58 back_ptr[t, k] = j;59 best_logp[t, k] = logp;60 }61 }62 }63 }64 log_p_z_star = max(best_logp[N]);65 for (k in 1 : K) {66 if (best_logp[N, k] == log_p_z_star) {67 z_star[N] = k;68 }69 }70 for (t in 1 : (N - 1)) {71 z_star[N - t] = back_ptr[N - t + 1, z_star[N - t + 1]];72 }73 }74}7576//@ DATA { N: 100, K: 2, y: [100 values] } // values supplied at runtime77//@ PARAMS ["theta1[1]","theta1[2]","theta2[1]","theta2[2]","mu[1]","mu[2]"]78//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}79
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0092 ≤ tol 0.0514 · floors 0.0257/0.0135 |
For each of N measurement time points, observations are provided: the time ts[n] and measured populations y[n,1] (prey) and y[n,2] (predator). Initial measured populations y_init[1] (prey) and y_init[2] (predator) are also observed, all positive real numbers. The model includes four positive system parameters theta[1] (prey birth rate), theta[2] (predation rate on prey), theta[3] (predator death rate), and theta[4] (predator efficiency); two positive initial populations z_init[1] (prey) and z_init[2] (predator) at time 0; and two positive measurement error standard deviations sigma[1] (prey) and sigma[2] (predator). The priors are: theta[1] and theta[3] each have a normal distribution with mean 1 and standard deviation 0.5; theta[2] and theta[4] each have a normal distribution with mean 0.05 and standard deviation 0.05; z_init[1] and z_init[2] each have a lognormal distribution with location (log-scale mean) log(10) and scale 1; sigma[1] and sigma[2] each have a lognormal distribution with location -1 and scale 1.
The latent prey and predator populations evolve according to the Lotka-Volterra differential equations. Denoting prey as u and predator as v, the system is du/dt = (theta[1] - theta[2]*v)*u and dv/dt = (-theta[3] + theta[4]*u)*v, with initial conditions u(0) = z_init[1] and v(0) = z_init[2]. The system is integrated numerically at each of the N measurement times to yield latent population trajectories. The observed initial populations y_init[1] and y_init[2] are independent lognormal random variables: y_init[k] has distribution lognormal with location log(z_init[k]) and scale sigma[k] for k = 1, 2. At each measurement time n, the observed populations y[n,1] and y[n,2] are independent, with y[n,k] distributed as lognormal with location log of the corresponding latent population and scale sigma[k].
The marginal posterior distributions of the eight parameters: theta[1] (prey birth rate), theta[2] (predation rate on prey), theta[3] (predator death rate), theta[4] (predator efficiency), z_init[1] (initial prey population), z_init[2] (initial predator population), sigma[1] (measurement error standard deviation for prey), and sigma[2] (measurement error standard deviation for predator).
answer spec
{
"kind": "record",
"fields": {
"theta[1]": {
"kind": "dist",
"domain": "real"
},
"theta[2]": {
"kind": "dist",
"domain": "real"
},
"theta[3]": {
"kind": "dist",
"domain": "real"
},
"theta[4]": {
"kind": "dist",
"domain": "real"
},
"z_init[1]": {
"kind": "dist",
"domain": "real"
},
"z_init[2]": {
"kind": "dist",
"domain": "real"
},
"sigma[1]": {
"kind": "dist",
"domain": "real"
},
"sigma[2]": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1functions {2 array[] real dz_dt(real t, // time3 array[] real z,4 // system state {prey, predator}5 array[] real theta, // parameters6 array[] real x_r, // unused data7 array[] int x_i) {8 real u = z[1];9 real v = z[2];1011 real alpha = theta[1];12 real beta = theta[2];13 real gamma = theta[3];14 real delta = theta[4];1516 real du_dt = (alpha - beta * v) * u;17 real dv_dt = (-gamma + delta * u) * v;18 return {du_dt, dv_dt};19 }20}21data {22 int<lower=0> N; // number of measurement times23 array[N] real ts; // measurement times > 024 array[2] real y_init; // initial measured populations25 array[N, 2] real<lower=0> y; // measured populations26}27parameters {28 array[4] real<lower=0> theta; // { alpha, beta, gamma, delta }29 array[2] real<lower=0> z_init; // initial population30 array[2] real<lower=0> sigma; // measurement errors31}32transformed parameters {33 array[N, 2] real z = integrate_ode_rk45(dz_dt, z_init, 0, ts, theta,34 rep_array(0.0, 0), rep_array(35 0, 0), 1e-5, 1e-3, 5e2);36}37model {38 theta[{1, 3}] ~ normal(1, 0.5);39 theta[{2, 4}] ~ normal(0.05, 0.05);40 sigma ~ lognormal(-1, 1);41 z_init ~ lognormal(log(10), 1);42 for (k in 1 : 2) {43 y_init[k] ~ lognormal(log(z_init[k]), sigma[k]);44 y[ : , k] ~ lognormal(log(z[ : , k]), sigma[k]);45 }46}47generated quantities {48 array[2] real y_init_rep;49 array[N, 2] real y_rep;50 for (k in 1 : 2) {51 y_init_rep[k] = lognormal_rng(log(z_init[k]), sigma[k]);52 for (n in 1 : N) {53 y_rep[n, k] = lognormal_rng(log(z[n, k]), sigma[k]);54 }55 }56}5758//@ DATA { N: 20, ts: [20 values], y_init: [2 values], y: [20×2 matrix] } // values supplied at runtime59//@ PARAMS ["theta[1]","theta[2]","theta[3]","theta[4]","z_init[1]","z_init[2]","sigma[1]","sigma[2]"]60//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}61
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.1012 ≤ tol 0.4363 · floors 0.2181/0.1800 |
Item-response data for I = 20 items and J = 100 students are provided as a binary matrix y[i,j] (1 = correct, 0 = incorrect), where i indexes items and j indexes students. The student ability standard deviation sigma_theta is constrained positive with a half-Cauchy(location 0, scale 2) prior. The item discrimination standard deviation sigma_a is constrained positive with a half-Cauchy(location 0, scale 2) prior. The item difficulty mean mu_b has a Normal(mean 0, sd 5) prior. The item difficulty standard deviation sigma_b is constrained positive with a half-Cauchy(location 0, scale 2) prior.
Each student j has a latent ability theta_j drawn from Normal(0, sigma_theta). Each item i has a discrimination parameter a_i drawn from LogNormal(0, sigma_a) (constrained positive) and a difficulty parameter b_i drawn from Normal(mu_b, sigma_b). The probability that student j answers item i correctly is inv_logit(a_i * (theta_j - b_i)).
The marginal posterior distribution of each parameter given the data: the ability standard deviation sigma_theta, the J = 100 student abilities theta_1, ..., theta_100, the discrimination standard deviation sigma_a, the I = 20 item discriminations a_1, ..., a_20, the difficulty mean mu_b, the difficulty standard deviation sigma_b, and the I = 20 item difficulties b_1, ..., b_20.
answer spec
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> I;3 int<lower=0> J;4 array[I, J] int<lower=0, upper=1> y;5}6parameters {7 real<lower=0> sigma_theta;8 vector[J] theta;910 real<lower=0> sigma_a;11 vector<lower=0>[I] a;1213 real mu_b;14 real<lower=0> sigma_b;15 vector[I] b;16}17model {18 sigma_theta ~ cauchy(0, 2);19 theta ~ normal(0, sigma_theta);2021 sigma_a ~ cauchy(0, 2);22 a ~ lognormal(0, sigma_a);2324 mu_b ~ normal(0, 5);25 sigma_b ~ cauchy(0, 2);26 b ~ normal(mu_b, sigma_b);2728 for (i in 1 : I) {29 y[i] ~ bernoulli_logit(a[i] * (theta - b[i]));30 }31}3233//@ DATA { I: 20, J: 100, y: [20×100 matrix] } // values supplied at runtime34//@ PARAMS ["sigma_theta","theta[1]","theta[2]","theta[3]","theta[4]","theta[5]","theta[6]","theta[7]","theta[8]","theta[9]","theta[10]","theta[11]","theta[12]","theta[13]","theta[14]","theta[15]","theta[16]","theta[17]","theta[18]","theta[19]","theta[20]","theta[21]","theta[22]","theta[23]","theta[24]","theta[25]","theta[26]","theta[27]","theta[28]","theta[29]","theta[30]","theta[31]","theta[32]","theta[33]","theta[34]","theta[35]","theta[36]","theta[37]","theta[38]","theta[39]","theta[40]","theta[41]","theta[42]","theta[43]","theta[44]","theta[45]","theta[46]","theta[47]","theta[48]","theta[49]","theta[50]","theta[51]","theta[52]","theta[53]","theta[54]","theta[55]","theta[56]","theta[57]","theta[58]","theta[59]","theta[60]","theta[61]","theta[62]","theta[63]","theta[64]","theta[65]","theta[66]","theta[67]","theta[68]","theta[69]","theta[70]","theta[71]","theta[72]","theta[73]","theta[74]","theta[75]","theta[76]","theta[77]","theta[78]","theta[79]","theta[80]","theta[81]","theta[82]","theta[83]","theta[84]","theta[85]","theta[86]","theta[87]","theta[88]","theta[89]","theta[90]","theta[91]","theta[92]","theta[93]","theta[94]","theta[95]","theta[96]","theta[97]","theta[98]","theta[99]","theta[100]","sigma_a","a[1]","a[2]","a[3]","a[4]","a[5]","a[6]","a[7]","a[8]","a[9]","a[10]","a[11]","a[12]","a[13]","a[14]","a[15]","a[16]","a[17]","a[18]","a[19]","a[20]","mu_b","sigma_b","b[1]","b[2]","b[3]","b[4]","b[5]","b[6]","b[7]","b[8]","b[9]","b[10]","b[11]","b[12]","b[13]","b[14]","b[15]","b[16]","b[17]","b[18]","b[19]","b[20]"]35//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}36
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| sigma_theta | 1.04 ± 0.191 | — |
| theta[1] | -0.670 ± 0.518 | — |
| theta[2] | 0.243 ± 0.547 | — |
| theta[3] | 1.06 ± 0.604 | — |
| theta[4] | -0.737 ± 0.547 | — |
| theta[5] | -0.710 ± 0.509 | — |
| theta[6] | -0.918 ± 0.544 | — |
| theta[7] | -0.256 ± 0.533 | — |
| theta[8] | 1.42 ± 0.658 | — |
| theta[9] | 0.386 ± 0.568 | — |
| theta[10] | 1.66 ± 0.733 | — |
| theta[11] | -0.195 ± 0.537 | — |
| theta[12] | -0.836 ± 0.536 | — |
| theta[13] | -0.617 ± 0.534 | — |
| theta[14] | 0.221 ± 0.560 | — |
| theta[15] | -1.76 ± 0.561 | — |
| theta[16] | -0.459 ± 0.550 | — |
| theta[17] | 1.80 ± 0.759 | — |
| theta[18] | 0.227 ± 0.566 | — |
| theta[19] | -0.183 ± 0.523 | — |
| theta[20] | -1.04 ± 0.541 | — |
| theta[21] | 0.471 ± 0.579 | — |
| theta[22] | 1.03 ± 0.618 | — |
| theta[23] | 0.156 ± 0.547 | — |
| theta[24] | 2.08 ± 0.743 | — |
| theta[25] | 0.976 ± 0.639 | — |
| theta[26] | 1.33 ± 0.644 | — |
| theta[27] | 0.312 ± 0.570 | — |
| theta[28] | 0.592 ± 0.599 | — |
| theta[29] | 0.956 ± 0.615 | — |
| theta[30] | 0.384 ± 0.546 | — |
| theta[31] | -0.275 ± 0.505 | — |
| theta[32] | 0.949 ± 0.576 | — |
| theta[33] | -0.031 ± 0.544 | — |
| theta[34] | -0.944 ± 0.502 | — |
| theta[35] | -0.761 ± 0.532 | — |
| theta[36] | 0.329 ± 0.541 | — |
| theta[37] | -1.90 ± 0.593 | — |
| theta[38] | 0.691 ± 0.630 | — |
| theta[39] | 0.167 ± 0.581 | — |
| theta[40] | -0.294 ± 0.498 | — |
| theta[41] | 1.04 ± 0.628 | — |
| theta[42] | -0.971 ± 0.517 | — |
| theta[43] | -0.511 ± 0.493 | — |
| theta[44] | -0.230 ± 0.559 | — |
| theta[45] | 0.275 ± 0.573 | — |
| theta[46] | 0.746 ± 0.634 | — |
| theta[47] | 1.70 ± 0.719 | — |
| theta[48] | 0.514 ± 0.591 | — |
| theta[49] | -0.395 ± 0.518 | — |
| theta[50] | -0.129 ± 0.514 | — |
| theta[51] | -0.976 ± 0.537 | — |
| theta[52] | -1.06 ± 0.509 | — |
| theta[53] | 0.476 ± 0.590 | — |
| theta[54] | -0.525 ± 0.557 | — |
| theta[55] | -0.992 ± 0.538 | — |
| theta[56] | 0.443 ± 0.562 | — |
| theta[57] | 0.718 ± 0.638 | — |
| theta[58] | 0.333 ± 0.588 | — |
| theta[59] | 1.61 ± 0.690 | — |
| theta[60] | 0.085 ± 0.559 | — |
| theta[61] | 0.013 ± 0.592 | — |
| theta[62] | -0.821 ± 0.547 | — |
| theta[63] | 0.314 ± 0.560 | — |
| theta[64] | -1.30 ± 0.512 | — |
| theta[65] | -1.33 ± 0.534 | — |
| theta[66] | 0.318 ± 0.582 | — |
| theta[67] | 0.519 ± 0.606 | — |
| theta[68] | -0.147 ± 0.556 | — |
| theta[69] | -0.316 ± 0.512 | — |
| theta[70] | -0.574 ± 0.487 | — |
| theta[71] | -0.298 ± 0.525 | — |
| theta[72] | -1.68 ± 0.556 | — |
| theta[73] | -1.14 ± 0.535 | — |
| theta[74] | 0.326 ± 0.528 | — |
| theta[75] | 0.660 ± 0.608 | — |
| theta[76] | 0.980 ± 0.612 | — |
| theta[77] | 0.085 ± 0.541 | — |
| theta[78] | -0.638 ± 0.536 | — |
| theta[79] | -1.38 ± 0.564 | — |
| theta[80] | -0.226 ± 0.518 | — |
| theta[81] | 0.365 ± 0.577 | — |
| theta[82] | -0.255 ± 0.554 | — |
| theta[83] | 1.41 ± 0.641 | — |
| theta[84] | 0.960 ± 0.623 | — |
| theta[85] | 0.860 ± 0.638 | — |
| theta[86] | -1.53 ± 0.570 | — |
| theta[87] | -0.319 ± 0.549 | — |
| theta[88] | -0.195 ± 0.540 | — |
| theta[89] | -0.540 ± 0.547 | — |
| theta[90] | -1.47 ± 0.568 | — |
| theta[91] | -0.949 ± 0.561 | — |
| theta[92] | 0.970 ± 0.587 | — |
| theta[93] | 0.103 ± 0.545 | — |
| theta[94] | -0.236 ± 0.531 | — |
| theta[95] | -0.327 ± 0.561 | — |
| theta[96] | 1.49 ± 0.725 | — |
| theta[97] | 0.253 ± 0.527 | — |
| theta[98] | 0.325 ± 0.548 | — |
| theta[99] | -0.350 ± 0.549 | — |
| theta[100] | -1.08 ± 0.523 | — |
| sigma_a | 0.516 ± 0.179 | — |
| a[1] | 1.08 ± 0.376 | — |
| a[2] | 1.29 ± 0.443 | — |
| a[3] | 1.03 ± 0.337 | — |
| a[4] | 0.866 ± 0.276 | — |
| a[5] | 1.13 ± 0.404 | — |
| a[6] | 1.10 ± 0.382 | — |
| a[7] | 0.902 ± 0.286 | — |
| a[8] | 0.470 ± 0.194 | — |
| a[9] | 1.96 ± 1.39 | — |
| a[10] | 0.832 ± 0.266 | — |
| a[11] | 1.46 ± 0.491 | — |
| a[12] | 1.86 ± 1.14 | — |
| a[13] | 0.954 ± 0.305 | — |
| a[14] | 1.44 ± 0.503 | — |
| a[15] | 1.04 ± 0.360 | — |
| a[16] | 1.04 ± 0.328 | — |
| a[17] | 0.786 ± 0.269 | — |
| a[18] | 1.67 ± 0.615 | — |
| a[19] | 0.721 ± 0.252 | — |
| a[20] | 0.540 ± 0.194 | — |
| mu_b | -0.980 ± 0.557 | — |
| sigma_b | 2.20 ± 0.539 | — |
| b[1] | -3.69 ± 1.12 | — |
| b[2] | 2.02 ± 0.559 | — |
| b[3] | 0.134 ± 0.272 | — |
| b[4] | 0.383 ± 0.342 | — |
| b[5] | -3.33 ± 0.970 | — |
| b[6] | -2.37 ± 0.738 | — |
| b[7] | -3.15 ± 1.01 | — |
| b[8] | -1.56 ± 0.776 | — |
| b[9] | -4.75 ± 1.42 | — |
| b[10] | 2.00 ± 0.688 | — |
| b[11] | -1.12 ± 0.341 | — |
| b[12] | -2.33 ± 0.672 | — |
| b[13] | 2.79 ± 0.804 | — |
| b[14] | 0.023 ± 0.212 | — |
| b[15] | -1.50 ± 0.533 | — |
| b[16] | -0.603 ± 0.291 | — |
| b[17] | -0.937 ± 0.492 | — |
| b[18] | -1.51 ± 0.410 | — |
| b[19] | -1.28 ± 0.605 | — |
| b[20] | 1.56 ± 0.750 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0745 ≤ tol 0.3174 · floors 0.1587/0.1160 |
For each of N = 434 children, the data provide the child's cognitive test score (bounded between 0 and 200), a binary indicator of whether the child's mother completed high school (1 = yes, 0 = no), and the child's mother's IQ score (bounded between 0 and 200). The model includes four regression coefficients—an intercept and slopes on the mother's high school completion, the mother's IQ, and the interaction between these two predictors—each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a half-Cauchy(location 0, scale 2.5) prior.
Each child's cognitive test score is normally distributed with a mean equal to the intercept plus the slope on mother's high school completion times the indicator, plus the slope on mother's IQ times the IQ score, plus the slope on the interaction between these two predictors times their product, and a common standard deviation sigma across all children.
The marginal posterior distributions of the five parameters: the intercept (reported as beta[1]), the slope on the mother's high school indicator (reported as beta[2]), the slope on the mother's IQ (reported as beta[3]), the slope on the interaction (reported as beta[4]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector<lower=0, upper=200>[N] kid_score;4 vector<lower=0, upper=200>[N] mom_iq;5 vector<lower=0, upper=1>[N] mom_hs;6}7transformed data {8 // interaction9 vector[N] inter;10 inter = mom_hs .* mom_iq;11}12parameters {13 vector[4] beta;14 real<lower=0> sigma;15}16model {17 sigma ~ cauchy(0, 2.5);18 kid_score ~ normal(beta[1] + beta[2] * mom_hs + beta[3] * mom_iq19 + beta[4] * inter, sigma);20}2122//@ DATA { N: 434, kid_score: [434 values], mom_hs: [434 values], mom_iq: [434 values] } // values supplied at runtime23//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","sigma"]24//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}25
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.5886 ≤ tol 2.9477 · floors 1.0106/1.4738 |
For each of N = 434 children the data provide the child's cognitive test score (between 0 and 200) and a binary indicator of whether the child's mother completed high school (1 = yes, 0 = no). The regression has two coefficients, an intercept and a slope on the high-school indicator, each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a half-Cauchy(location 0, scale 2.5) prior.
Each child's test score is Normal-distributed with a mean equal to the intercept plus the slope times the mother's high-school indicator, and a common standard deviation sigma across all children.
The marginal posterior distribution of each of the three parameters: the intercept (reported as beta[1]), the slope on the mother's high-school indicator (reported as beta[2]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector<lower=0, upper=200>[N] kid_score;4 vector<lower=0, upper=1>[N] mom_hs;5}6parameters {7 vector[2] beta;8 real<lower=0> sigma;9}10model {11 sigma ~ cauchy(0, 2.5);12 kid_score ~ normal(beta[1] + beta[2] * mom_hs, sigma);13}1415//@ DATA { N: 434, kid_score: [434 values], mom_hs: [434 values] } // values supplied at runtime16//@ PARAMS ["beta[1]","beta[2]","sigma"]17//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}18
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0842 ≤ tol 0.2948 · floors 0.1474/0.1196 |
For each of N = 434 children, the data provide the child's cognitive test score (between 0 and 200), an indicator of whether the mother completed high school (1 = yes, 0 = no), and the mother's cognitive test score. The regression has three coefficients: an intercept and slopes for the high-school indicator and the mother's score, each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a half-Cauchy(location 0, scale 2.5) prior.
Each child's test score is Normal-distributed with a mean equal to the intercept plus the mother's high-school indicator coefficient times the indicator, plus the mother's IQ coefficient times the mother's score, and a common standard deviation sigma across all children.
The marginal posterior distribution of each of the four parameters: the intercept (reported as beta[1]), the mother's high school completion effect (reported as beta[2]), the mother's IQ effect (reported as beta[3]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector<lower=0, upper=200>[N] kid_score;4 vector<lower=0, upper=200>[N] mom_iq;5 vector<lower=0, upper=1>[N] mom_hs;6}7parameters {8 vector[3] beta;9 real<lower=0> sigma;10}11model {12 sigma ~ cauchy(0, 2.5);13 kid_score ~ normal(beta[1] + beta[2] * mom_hs + beta[3] * mom_iq, sigma);14}1516//@ DATA { N: 434, kid_score: [434 values], mom_hs: [434 values], mom_iq: [434 values] } // values supplied at runtime17//@ PARAMS ["beta[1]","beta[2]","beta[3]","sigma"]18//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}19
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.2133 ≤ tol 1.2171 · floors 0.6085/0.4829 |
For each of N = 434 children the data provide the child's intelligence quotient (IQ) test score (between 0 and 200) and the child's mother's IQ score (between 0 and 200). The regression has two coefficients, an intercept and a slope on maternal IQ, each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a half-Cauchy(location 0, scale 2.5) prior.
Each child's test score is Normal-distributed with a mean equal to the intercept plus the slope times the mother's IQ score, and a common standard deviation sigma across all children.
The marginal posterior distribution of each of the three parameters: the intercept (reported as beta[1]), the slope on maternal IQ (reported as beta[2]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector<lower=0, upper=200>[N] kid_score;4 vector<lower=0, upper=200>[N] mom_iq;5}6parameters {7 vector[2] beta;8 real<lower=0> sigma;9}10model {11 sigma ~ cauchy(0, 2.5);12 kid_score ~ normal(beta[1] + beta[2] * mom_iq, sigma);13}1415//@ DATA { N: 434, kid_score: [434 values], mom_iq: [434 values] } // values supplied at runtime16//@ PARAMS ["beta[1]","beta[2]","sigma"]17//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}18
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.3178 ≤ tol 0.7203 · floors 0.3601/0.3502 |
For each of N = 434 children, the data provide the child's cognitive test score and two parent measures: the mother's high school completion status and the mother's IQ score. To facilitate interpretation, the model centers both the mother's high school measure and the mother's IQ by subtracting their sample means. The regression includes four coefficients: an intercept, a slope for centered mother's high school, a slope for centered mother's IQ, and a slope for their interaction. The regression coefficients beta[1], beta[2], beta[3], and beta[4] each have a flat (improper uniform) prior over the reals. The error standard deviation sigma, constrained to be positive, has a half-Cauchy(location 0, scale 2.5) prior.
Each child's test score is normally distributed with a mean equal to the intercept plus the slope for centered mother's high school times that child's centered mother's high school value, plus the slope for centered mother's IQ times that child's centered mother's IQ value, plus the interaction coefficient times the product of the centered mother's high school and centered mother's IQ values. The standard deviation of this normal distribution is sigma, shared across all children.
The marginal posterior distribution of each of the five parameters: the intercept (reported as beta[1]), the coefficient for centered mother's high school measure (reported as beta[2]), the coefficient for centered mother's IQ (reported as beta[3]), the coefficient for the interaction between centered mother's high school measure and centered mother's IQ (reported as beta[4]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] kid_score;4 vector[N] mom_hs;5 vector[N] mom_iq;6}7transformed data {8 // centered predictors9 vector[N] c_mom_hs;10 vector[N] c_mom_iq;11 vector[N] inter;12 c_mom_hs = mom_hs - mean(mom_hs);13 c_mom_iq = mom_iq - mean(mom_iq);14 inter = c_mom_hs .* c_mom_iq;15}16parameters {17 vector[4] beta;18 real<lower=0> sigma;19}20model {21 kid_score ~ normal(beta[1] + beta[2] * c_mom_hs + beta[3] * c_mom_iq22 + beta[4] * inter, sigma);23}2425//@ DATA { N: 434, kid_score: [434 values], mom_hs: [434 values], mom_iq: [434 values] } // values supplied at runtime26//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","sigma"]27//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}28
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0660 ≤ tol 0.2432 · floors 0.0958/0.1141 |
For each of N = 434 children, the data provide the child's cognitive test score and two maternal characteristics: whether the child's mother completed high school (1 = yes, 0 = no) and the mother's IQ score. The regression has four coefficients. Two are defined on centered versions of the maternal predictors: maternal high school completion is centered at 0.5, and maternal IQ is centered at 100. The third coefficient is the intercept. The fourth is the slope on the interaction between the two centered predictors. Each of the four coefficients has a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each child's test score is normally distributed with mean equal to the intercept plus a linear combination of the centered maternal predictors and their interaction, with a common standard deviation sigma. Specifically, the centered maternal high school indicator is computed as mom_hs minus 0.5, the centered maternal IQ is computed as mom_iq minus 100, and the interaction is the product of these two centered values. The mean is then beta[1] (intercept) plus beta[2] times the centered high school indicator, plus beta[3] times the centered IQ, plus beta[4] times the interaction, with standard deviation sigma.
The marginal posterior distributions of the five parameters: the intercept beta[1], the coefficient on centered maternal high school completion beta[2], the coefficient on centered maternal IQ beta[3], the coefficient on the interaction between centered maternal high school and centered maternal IQ beta[4], and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] kid_score;4 vector[N] mom_hs;5 vector[N] mom_iq;6}7transformed data {8 // centering on reference points9 vector[N] c2_mom_hs;10 vector[N] c2_mom_iq;11 vector[N] inter;12 c2_mom_hs = mom_hs - 0.5;13 c2_mom_iq = mom_iq - 100;14 inter = c2_mom_hs .* c2_mom_iq;15}16parameters {17 vector[4] beta;18 real<lower=0> sigma;19}20model {21 kid_score ~ normal(beta[1] + beta[2] * c2_mom_hs + beta[3] * c2_mom_iq22 + beta[4] * inter, sigma);23}2425//@ DATA { N: 434, kid_score: [434 values], mom_hs: [434 values], mom_iq: [434 values] } // values supplied at runtime26//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","sigma"]27//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}28
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0875 ≤ tol 0.2602 · floors 0.1082/0.1038 |
For each of N children, the data provide the child's cognitive test score, a binary indicator of whether the mother completed high school, and the mother's IQ score. The regression has four coefficients: an intercept, a slope on the standardized mother's high school indicator, a slope on the standardized mother's IQ, and a slope on the interaction between these two standardized predictors. Standardization of each predictor is performed by subtracting the sample mean and dividing by twice the sample standard deviation. The four regression coefficients (beta[1], beta[2], beta[3], beta[4]) each have a flat (improper uniform) prior over the reals. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each child's test score is normally distributed with a mean equal to the intercept plus the slope on standardized mother's high school indicator times that indicator, plus the slope on standardized mother's IQ times the mother's IQ, plus the slope on the interaction times the product of the two standardized predictors, and a common standard deviation sigma across all children.
The marginal posterior distributions of each of the five parameters: the intercept (reported as beta[1]), the slope on the standardized mother's high school indicator (reported as beta[2]), the slope on the standardized mother's IQ (reported as beta[3]), the slope on the interaction between standardized predictors (reported as beta[4]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] kid_score;4 vector[N] mom_hs;5 vector[N] mom_iq;6}7transformed data {8 // standardizing9 vector[N] z_mom_hs;10 vector[N] z_mom_iq;11 vector[N] inter;12 z_mom_hs = (mom_hs - mean(mom_hs)) / (2 * sd(mom_hs));13 z_mom_iq = (mom_iq - mean(mom_iq)) / (2 * sd(mom_iq));14 inter = z_mom_hs .* z_mom_iq;15}16parameters {17 vector[4] beta;18 real<lower=0> sigma;19}20model {21 kid_score ~ normal(beta[1] + beta[2] * z_mom_hs + beta[3] * z_mom_iq22 + beta[4] * inter, sigma);23}2425//@ DATA { N: 434, kid_score: [434 values], mom_hs: [434 values], mom_iq: [434 values] } // values supplied at runtime26//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","sigma"]27//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}28
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.1480 ≤ tol 0.8464 · floors 0.2620/0.2591 |
For each of N = 434 children the data provide the child's cognitive test score and the mother's employment status, encoded as an integer category with four levels. The regression has four coefficients, one for each employment status category, each with a flat (improper uniform) prior over the reals. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each child's test score is normally distributed with a mean determined by the mother's employment status category and a common standard deviation sigma across all children. Specifically, the mean is given by the coefficient corresponding to the child's mother's employment status category.
The marginal posterior distribution of each of the five parameters: beta[1], beta[2], beta[3], and beta[4] (the four employment-status-specific mean parameters), and sigma (the error standard deviation).
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] kid_score;4 array[N] int mom_work;5}6transformed data {7 vector[N] work2;8 vector[N] work3;9 vector[N] work4;10 for (i in 1 : N) {11 work2[i] = mom_work[i] == 2;12 work3[i] = mom_work[i] == 3;13 work4[i] = mom_work[i] == 4;14 }15}16parameters {17 vector[4] beta;18 real<lower=0> sigma;19}20model {21 kid_score ~ normal(beta[1] + beta[2] * work2 + beta[3] * work322 + beta[4] * work4, sigma);23}2425//@ DATA { N: 434, kid_score: [434 values], mom_work: [434 values] } // values supplied at runtime26//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","sigma"]27//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}28
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.1269 ≤ tol 0.4941 · floors 0.2195/0.2470 |
The data comprise N observations, each consisting of a predictor value x and a response value y. Additionally, the data supply four hyperparameters that specify the prior distributions: pmualpha and psalpha are the mean and standard deviation of the prior for the intercept alpha; pmubeta and psbeta are the mean and standard deviation of the prior for the slope beta. The intercept alpha has a Normal(pmualpha, psalpha) prior. The slope beta has a Normal(pmubeta, psbeta) prior. The error standard deviation sigma, constrained positive, has an improper flat (uniform) prior over the positive reals.
Each observation's response value y is Normal-distributed with a mean equal to the intercept alpha plus the slope beta times that observation's predictor value x, and a common standard deviation sigma across all observations.
The marginal posterior distribution of each of the three parameters: alpha (the intercept), beta (the slope), and sigma (the error standard deviation).
answer spec
{
"kind": "record",
"fields": {
"alpha": {
"kind": "dist",
"domain": "real"
},
"beta": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// Gaussian linear model with adjustable priors2data {3 int<lower=0> N; // number of data points4 vector[N] x; //5 vector[N] y; //6 real xpred; // input location for prediction7 real pmualpha; // prior mean for alpha8 real psalpha; // prior std for alpha9 real pmubeta; // prior mean for beta10 real psbeta; // prior std for beta11}12parameters {13 real alpha;14 real beta;15 real<lower=0> sigma;16}17model {18 alpha ~ normal(pmualpha, psalpha);19 beta ~ normal(pmubeta, psbeta);20 y ~ normal(alpha + beta * x, sigma);21}2223//@ DATA { N: 62, x: [62 values], y: [62 values], xpred: 2016, pmualpha: 9.31290322580645, psalpha: 100, pmubeta: 0, psbeta: 0.0333333333333333 } // values supplied at runtime24//@ PARAMS ["alpha","beta","sigma"]25//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}26
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=1.4862 ≤ tol 5.6980 · floors 2.8490/1.9225 |
For each of N observations the data provide a single real value y. The model infers a two-component Gaussian mixture in which the two component means are ordered (mu[1] is less than or equal to mu[2]), and each component has its own positive standard deviation. The means mu[1] and mu[2], with the ordering constraint, have a Normal(0, 2) prior. Each standard deviation sigma[1] and sigma[2] has an independent Normal(0, 2) prior, constrained positive. The mixing weight theta, the probability of component 1, has a Beta(5, 5) prior and is constrained to lie in [0, 1].
A two-component Gaussian mixture model generates the observed data. Each observation is generated from one of two components: with probability theta it is drawn from component 1 (a Normal distribution with mean mu[1] and standard deviation sigma[1]), and with probability 1 - theta it is drawn from component 2 (a Normal distribution with mean mu[2] and standard deviation sigma[2]). The two component means satisfy an ordering: mu[1] is less than or equal to mu[2]. All observations are independent draws from this mixture distribution.
The marginal posterior distributions of the five parameters: mu[1] and mu[2] (the ordered component means), sigma[1] and sigma[2] (the component standard deviations), and theta (the mixing weight for component 1).
answer spec
{
"kind": "record",
"fields": {
"mu[1]": {
"kind": "dist",
"domain": "real"
},
"mu[2]": {
"kind": "dist",
"domain": "real"
},
"sigma[1]": {
"kind": "dist",
"domain": "real"
},
"sigma[2]": {
"kind": "dist",
"domain": "real"
},
"theta": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] y;4}5parameters {6 ordered[2] mu;7 array[2] real<lower=0> sigma;8 real<lower=0, upper=1> theta;9}10model {11 sigma ~ normal(0, 2);12 mu ~ normal(0, 2);13 theta ~ beta(5, 5);14 for (n in 1 : N) {15 target += log_mix(theta, normal_lpdf(y[n] | mu[1], sigma[1]),16 normal_lpdf(y[n] | mu[2], sigma[2]));17 }18}1920//@ DATA { N: 1000, y: [1000 values] } // values supplied at runtime21//@ PARAMS ["mu[1]","mu[2]","sigma[1]","sigma[2]","theta"]22//@ SAMPLING {"chains":8,"iter_warmup":6000,"iter_sampling":3000,"adapt_delta":0.8}23
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0015 ≤ tol 0.0069 · floors 0.0008/0.0034 |
Responses from N = 1000 students to T = 5 LSAT questions are provided. The data are encoded as R = 32 response patterns (all possible binary vectors of length 5), each with a cumulative count indicating how many of the 1000 students exhibit that pattern; the full individual response matrix is reconstructed from these pattern counts. Each question k (k = 1..5) has a difficulty parameter alpha_k with a Normal(mean 0, sd 100) prior. Each student j (j = 1..1000) has an ability parameter theta_j with a Normal(mean 0, sd 1) prior. The common discrimination parameter beta has a Normal(mean 0, sd 100) prior.
Each student j has a latent ability theta_j. Each question k has a difficulty alpha_k. The probability that student j answers question k correctly is inv_logit(beta * theta_j - alpha_k). The observed binary response for student j on question k is Bernoulli with that probability, independently across students and questions.
The marginal posterior distribution of each parameter given the data: the five question difficulty parameters alpha_1, ..., alpha_5, the 1000 student ability parameters theta_1, ..., theta_1000, the common discrimination parameter beta, the posterior mean difficulty mean_alpha (= mean of alpha_1..alpha_5), and the five mean-centered difficulty parameters a_1, ..., a_5 (where a_k = alpha_k - mean_alpha).
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N; // 1000, number of students3 int<lower=0> R; // 32, number of patterns of results: 2^T4 int<lower=0> T; // 5, number of questions5 array[R] int<lower=0> culm;6 array[R, T] int<lower=0> response;7}8transformed data {9 array[T, N] int r;10 vector[N] ones;1112 for (j in 1 : culm[1]) {13 for (k in 1 : T) {14 r[k, j] = response[1, k];15 }16 }17 for (i in 2 : R) {18 for (j in (culm[i - 1] + 1) : culm[i]) {19 for (k in 1 : T) {20 r[k, j] = response[i, k];21 }22 }23 }24 for (i in 1 : N) {25 ones[i] = 1.0;26 }27}28parameters {29 array[T] real alpha;30 vector[N] theta;31 real<lower=0> beta;32}33model {34 alpha ~ normal(0, 100.);35 theta ~ normal(0, 1);36 beta ~ normal(0.0, 100.);37 for (k in 1 : T) {38 r[k] ~ bernoulli_logit(beta * theta - alpha[k] * ones);39 }40}41generated quantities {42 real mean_alpha;43 array[T] real a;44 mean_alpha = mean(alpha);45 for (t in 1 : T) {46 a[t] = alpha[t] - mean_alpha;47 }48}4950//@ DATA { response: [32×5 matrix], culm: [32 values], N: 1000, R: 32, T: 5 } // values supplied at runtime51//@ PARAMS ["alpha[1]","alpha[2]","alpha[3]","alpha[4]","alpha[5]","theta[1]","theta[2]","theta[3]","theta[4]","theta[5]","theta[6]","theta[7]","theta[8]","theta[9]","theta[10]","theta[11]","theta[12]","theta[13]","theta[14]","theta[15]","theta[16]","theta[17]","theta[18]","theta[19]","theta[20]","theta[21]","theta[22]","theta[23]","theta[24]","theta[25]","theta[26]","theta[27]","theta[28]","theta[29]","theta[30]","theta[31]","theta[32]","theta[33]","theta[34]","theta[35]","theta[36]","theta[37]","theta[38]","theta[39]","theta[40]","theta[41]","theta[42]","theta[43]","theta[44]","theta[45]","theta[46]","theta[47]","theta[48]","theta[49]","theta[50]","theta[51]","theta[52]","theta[53]","theta[54]","theta[55]","theta[56]","theta[57]","theta[58]","theta[59]","theta[60]","theta[61]","theta[62]","theta[63]","theta[64]","theta[65]","theta[66]","theta[67]","theta[68]","theta[69]","theta[70]","theta[71]","theta[72]","theta[73]","theta[74]","theta[75]","theta[76]","theta[77]","theta[78]","theta[79]","theta[80]","theta[81]","theta[82]","theta[83]","theta[84]","theta[85]","theta[86]","theta[87]","theta[88]","theta[89]","theta[90]","theta[91]","theta[92]","theta[93]","theta[94]","theta[95]","theta[96]","theta[97]","theta[98]","theta[99]","theta[100]","theta[101]","theta[102]","theta[103]","theta[104]","theta[105]","theta[106]","theta[107]","theta[108]","theta[109]","theta[110]","theta[111]","theta[112]","theta[113]","theta[114]","theta[115]","theta[116]","theta[117]","theta[118]","theta[119]","theta[120]","theta[121]","theta[122]","theta[123]","theta[124]","theta[125]","theta[126]","theta[127]","theta[128]","theta[129]","theta[130]","theta[131]","theta[132]","theta[133]","theta[134]","theta[135]","theta[136]","theta[137]","theta[138]","theta[139]","theta[140]","theta[141]","theta[142]","theta[143]","theta[144]","theta[145]","theta[146]","theta[147]","theta[148]","theta[149]","theta[150]","theta[151]","theta[152]","theta[153]","theta[154]","theta[155]","theta[156]","theta[157]","theta[158]","theta[159]","theta[160]","theta[161]","theta[162]","theta[163]","theta[164]","theta[165]","theta[166]","theta[167]","theta[168]","theta[169]","theta[170]","theta[171]","theta[172]","theta[173]","theta[174]","theta[175]","theta[176]","theta[177]","theta[178]","theta[179]","theta[180]","theta[181]","theta[182]","theta[183]","theta[184]","theta[185]","theta[186]","theta[187]","theta[188]","theta[189]","theta[190]","theta[191]","theta[192]","theta[193]","theta[194]","theta[195]","theta[196]","theta[197]","theta[198]","theta[199]","theta[200]","theta[201]","theta[202]","theta[203]","theta[204]","theta[205]","theta[206]","theta[207]","theta[208]","theta[209]","theta[210]","theta[211]","theta[212]","theta[213]","theta[214]","theta[215]","theta[216]","theta[217]","theta[218]","theta[219]","theta[220]","theta[221]","theta[222]","theta[223]","theta[224]","theta[225]","theta[226]","theta[227]","theta[228]","theta[229]","theta[230]","theta[231]","theta[232]","theta[233]","theta[234]","theta[235]","theta[236]","theta[237]","theta[238]","theta[239]","theta[240]","theta[241]","theta[242]","theta[243]","theta[244]","theta[245]","theta[246]","theta[247]","theta[248]","theta[249]","theta[250]","theta[251]","theta[252]","theta[253]","theta[254]","theta[255]","theta[256]","theta[257]","theta[258]","theta[259]","theta[260]","theta[261]","theta[262]","theta[263]","theta[264]","theta[265]","theta[266]","theta[267]","theta[268]","theta[269]","theta[270]","theta[271]","theta[272]","theta[273]","theta[274]","theta[275]","theta[276]","theta[277]","theta[278]","theta[279]","theta[280]","theta[281]","theta[282]","theta[283]","theta[284]","theta[285]","theta[286]","theta[287]","theta[288]","theta[289]","theta[290]","theta[291]","theta[292]","theta[293]","theta[294]","theta[295]","theta[296]","theta[297]","theta[298]","theta[299]","theta[300]","theta[301]","theta[302]","theta[303]","theta[304]","theta[305]","theta[306]","theta[307]","theta[308]","theta[309]","theta[310]","theta[311]","theta[312]","theta[313]","theta[314]","theta[315]","theta[316]","theta[317]","theta[318]","theta[319]","theta[320]","theta[321]","theta[322]","theta[323]","theta[324]","theta[325]","theta[326]","theta[327]","theta[328]","theta[329]","theta[330]","theta[331]","theta[332]","theta[333]","theta[334]","theta[335]","theta[336]","theta[337]","theta[338]","theta[339]","theta[340]","theta[341]","theta[342]","theta[343]","theta[344]","theta[345]","theta[346]","theta[347]","theta[348]","theta[349]","theta[350]","theta[351]","theta[352]","theta[353]","theta[354]","theta[355]","theta[356]","theta[357]","theta[358]","theta[359]","theta[360]","theta[361]","theta[362]","theta[363]","theta[364]","theta[365]","theta[366]","theta[367]","theta[368]","theta[369]","theta[370]","theta[371]","theta[372]","theta[373]","theta[374]","theta[375]","theta[376]","theta[377]","theta[378]","theta[379]","theta[380]","theta[381]","theta[382]","theta[383]","theta[384]","theta[385]","theta[386]","theta[387]","theta[388]","theta[389]","theta[390]","theta[391]","theta[392]","theta[393]","theta[394]","theta[395]","theta[396]","theta[397]","theta[398]","theta[399]","theta[400]","theta[401]","theta[402]","theta[403]","theta[404]","theta[405]","theta[406]","theta[407]","theta[408]","theta[409]","theta[410]","theta[411]","theta[412]","theta[413]","theta[414]","theta[415]","theta[416]","theta[417]","theta[418]","theta[419]","theta[420]","theta[421]","theta[422]","theta[423]","theta[424]","theta[425]","theta[426]","theta[427]","theta[428]","theta[429]","theta[430]","theta[431]","theta[432]","theta[433]","theta[434]","theta[435]","theta[436]","theta[437]","theta[438]","theta[439]","theta[440]","theta[441]","theta[442]","theta[443]","theta[444]","theta[445]","theta[446]","theta[447]","theta[448]","theta[449]","theta[450]","theta[451]","theta[452]","theta[453]","theta[454]","theta[455]","theta[456]","theta[457]","theta[458]","theta[459]","theta[460]","theta[461]","theta[462]","theta[463]","theta[464]","theta[465]","theta[466]","theta[467]","theta[468]","theta[469]","theta[470]","theta[471]","theta[472]","theta[473]","theta[474]","theta[475]","theta[476]","theta[477]","theta[478]","theta[479]","theta[480]","theta[481]","theta[482]","theta[483]","theta[484]","theta[485]","theta[486]","theta[487]","theta[488]","theta[489]","theta[490]","theta[491]","theta[492]","theta[493]","theta[494]","theta[495]","theta[496]","theta[497]","theta[498]","theta[499]","theta[500]","theta[501]","theta[502]","theta[503]","theta[504]","theta[505]","theta[506]","theta[507]","theta[508]","theta[509]","theta[510]","theta[511]","theta[512]","theta[513]","theta[514]","theta[515]","theta[516]","theta[517]","theta[518]","theta[519]","theta[520]","theta[521]","theta[522]","theta[523]","theta[524]","theta[525]","theta[526]","theta[527]","theta[528]","theta[529]","theta[530]","theta[531]","theta[532]","theta[533]","theta[534]","theta[535]","theta[536]","theta[537]","theta[538]","theta[539]","theta[540]","theta[541]","theta[542]","theta[543]","theta[544]","theta[545]","theta[546]","theta[547]","theta[548]","theta[549]","theta[550]","theta[551]","theta[552]","theta[553]","theta[554]","theta[555]","theta[556]","theta[557]","theta[558]","theta[559]","theta[560]","theta[561]","theta[562]","theta[563]","theta[564]","theta[565]","theta[566]","theta[567]","theta[568]","theta[569]","theta[570]","theta[571]","theta[572]","theta[573]","theta[574]","theta[575]","theta[576]","theta[577]","theta[578]","theta[579]","theta[580]","theta[581]","theta[582]","theta[583]","theta[584]","theta[585]","theta[586]","theta[587]","theta[588]","theta[589]","theta[590]","theta[591]","theta[592]","theta[593]","theta[594]","theta[595]","theta[596]","theta[597]","theta[598]","theta[599]","theta[600]","theta[601]","theta[602]","theta[603]","theta[604]","theta[605]","theta[606]","theta[607]","theta[608]","theta[609]","theta[610]","theta[611]","theta[612]","theta[613]","theta[614]","theta[615]","theta[616]","theta[617]","theta[618]","theta[619]","theta[620]","theta[621]","theta[622]","theta[623]","theta[624]","theta[625]","theta[626]","theta[627]","theta[628]","theta[629]","theta[630]","theta[631]","theta[632]","theta[633]","theta[634]","theta[635]","theta[636]","theta[637]","theta[638]","theta[639]","theta[640]","theta[641]","theta[642]","theta[643]","theta[644]","theta[645]","theta[646]","theta[647]","theta[648]","theta[649]","theta[650]","theta[651]","theta[652]","theta[653]","theta[654]","theta[655]","theta[656]","theta[657]","theta[658]","theta[659]","theta[660]","theta[661]","theta[662]","theta[663]","theta[664]","theta[665]","theta[666]","theta[667]","theta[668]","theta[669]","theta[670]","theta[671]","theta[672]","theta[673]","theta[674]","theta[675]","theta[676]","theta[677]","theta[678]","theta[679]","theta[680]","theta[681]","theta[682]","theta[683]","theta[684]","theta[685]","theta[686]","theta[687]","theta[688]","theta[689]","theta[690]","theta[691]","theta[692]","theta[693]","theta[694]","theta[695]","theta[696]","theta[697]","theta[698]","theta[699]","theta[700]","theta[701]","theta[702]","theta[703]","theta[704]","theta[705]","theta[706]","theta[707]","theta[708]","theta[709]","theta[710]","theta[711]","theta[712]","theta[713]","theta[714]","theta[715]","theta[716]","theta[717]","theta[718]","theta[719]","theta[720]","theta[721]","theta[722]","theta[723]","theta[724]","theta[725]","theta[726]","theta[727]","theta[728]","theta[729]","theta[730]","theta[731]","theta[732]","theta[733]","theta[734]","theta[735]","theta[736]","theta[737]","theta[738]","theta[739]","theta[740]","theta[741]","theta[742]","theta[743]","theta[744]","theta[745]","theta[746]","theta[747]","theta[748]","theta[749]","theta[750]","theta[751]","theta[752]","theta[753]","theta[754]","theta[755]","theta[756]","theta[757]","theta[758]","theta[759]","theta[760]","theta[761]","theta[762]","theta[763]","theta[764]","theta[765]","theta[766]","theta[767]","theta[768]","theta[769]","theta[770]","theta[771]","theta[772]","theta[773]","theta[774]","theta[775]","theta[776]","theta[777]","theta[778]","theta[779]","theta[780]","theta[781]","theta[782]","theta[783]","theta[784]","theta[785]","theta[786]","theta[787]","theta[788]","theta[789]","theta[790]","theta[791]","theta[792]","theta[793]","theta[794]","theta[795]","theta[796]","theta[797]","theta[798]","theta[799]","theta[800]","theta[801]","theta[802]","theta[803]","theta[804]","theta[805]","theta[806]","theta[807]","theta[808]","theta[809]","theta[810]","theta[811]","theta[812]","theta[813]","theta[814]","theta[815]","theta[816]","theta[817]","theta[818]","theta[819]","theta[820]","theta[821]","theta[822]","theta[823]","theta[824]","theta[825]","theta[826]","theta[827]","theta[828]","theta[829]","theta[830]","theta[831]","theta[832]","theta[833]","theta[834]","theta[835]","theta[836]","theta[837]","theta[838]","theta[839]","theta[840]","theta[841]","theta[842]","theta[843]","theta[844]","theta[845]","theta[846]","theta[847]","theta[848]","theta[849]","theta[850]","theta[851]","theta[852]","theta[853]","theta[854]","theta[855]","theta[856]","theta[857]","theta[858]","theta[859]","theta[860]","theta[861]","theta[862]","theta[863]","theta[864]","theta[865]","theta[866]","theta[867]","theta[868]","theta[869]","theta[870]","theta[871]","theta[872]","theta[873]","theta[874]","theta[875]","theta[876]","theta[877]","theta[878]","theta[879]","theta[880]","theta[881]","theta[882]","theta[883]","theta[884]","theta[885]","theta[886]","theta[887]","theta[888]","theta[889]","theta[890]","theta[891]","theta[892]","theta[893]","theta[894]","theta[895]","theta[896]","theta[897]","theta[898]","theta[899]","theta[900]","theta[901]","theta[902]","theta[903]","theta[904]","theta[905]","theta[906]","theta[907]","theta[908]","theta[909]","theta[910]","theta[911]","theta[912]","theta[913]","theta[914]","theta[915]","theta[916]","theta[917]","theta[918]","theta[919]","theta[920]","theta[921]","theta[922]","theta[923]","theta[924]","theta[925]","theta[926]","theta[927]","theta[928]","theta[929]","theta[930]","theta[931]","theta[932]","theta[933]","theta[934]","theta[935]","theta[936]","theta[937]","theta[938]","theta[939]","theta[940]","theta[941]","theta[942]","theta[943]","theta[944]","theta[945]","theta[946]","theta[947]","theta[948]","theta[949]","theta[950]","theta[951]","theta[952]","theta[953]","theta[954]","theta[955]","theta[956]","theta[957]","theta[958]","theta[959]","theta[960]","theta[961]","theta[962]","theta[963]","theta[964]","theta[965]","theta[966]","theta[967]","theta[968]","theta[969]","theta[970]","theta[971]","theta[972]","theta[973]","theta[974]","theta[975]","theta[976]","theta[977]","theta[978]","theta[979]","theta[980]","theta[981]","theta[982]","theta[983]","theta[984]","theta[985]","theta[986]","theta[987]","theta[988]","theta[989]","theta[990]","theta[991]","theta[992]","theta[993]","theta[994]","theta[995]","theta[996]","theta[997]","theta[998]","theta[999]","theta[1000]","beta","mean_alpha","a[1]","a[2]","a[3]","a[4]","a[5]"]52//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}53
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| alpha[1] | -2.74 ± 0.137 | — |
| alpha[2] | -1.00 ± 0.073 | — |
| alpha[3] | -0.239 ± 0.075 | — |
| alpha[4] | -1.31 ± 0.083 | — |
| alpha[5] | -2.10 ± 0.107 | — |
| theta[1] | -1.91 ± 0.761 | — |
| theta[2] | -1.92 ± 0.794 | — |
| theta[3] | -1.90 ± 0.812 | — |
| theta[4] | -1.44 ± 0.835 | — |
| theta[5] | -1.32 ± 0.771 | — |
| theta[6] | -1.38 ± 0.810 | — |
| theta[7] | -1.44 ± 0.819 | — |
| theta[8] | -1.36 ± 0.832 | — |
| theta[9] | -1.41 ± 0.780 | — |
| theta[10] | -1.42 ± 0.819 | — |
| theta[11] | -1.47 ± 0.798 | — |
| theta[12] | -0.881 ± 0.790 | — |
| theta[13] | -0.907 ± 0.804 | — |
| theta[14] | -0.868 ± 0.767 | — |
| theta[15] | -0.883 ± 0.852 | — |
| theta[16] | -0.898 ± 0.862 | — |
| theta[17] | -0.987 ± 0.837 | — |
| theta[18] | -0.931 ± 0.833 | — |
| theta[19] | -0.907 ± 0.875 | — |
| theta[20] | -0.989 ± 0.837 | — |
| theta[21] | -0.895 ± 0.791 | — |
| theta[22] | -0.941 ± 0.827 | — |
| theta[23] | -1.37 ± 0.844 | — |
| theta[24] | -0.955 ± 0.802 | — |
| theta[25] | -0.965 ± 0.822 | — |
| theta[26] | -0.959 ± 0.851 | — |
| theta[27] | -0.955 ± 0.785 | — |
| theta[28] | -0.446 ± 0.826 | — |
| theta[29] | -0.453 ± 0.806 | — |
| theta[30] | -0.395 ± 0.815 | — |
| theta[31] | -0.463 ± 0.783 | — |
| theta[32] | -1.39 ± 0.809 | — |
| theta[33] | -0.939 ± 0.792 | — |
| theta[34] | -0.963 ± 0.807 | — |
| theta[35] | -0.962 ± 0.802 | — |
| theta[36] | -0.898 ± 0.838 | — |
| theta[37] | -0.971 ± 0.797 | — |
| theta[38] | -0.960 ± 0.828 | — |
| theta[39] | -0.998 ± 0.821 | — |
| theta[40] | -0.894 ± 0.781 | — |
| theta[41] | -0.438 ± 0.788 | — |
| theta[42] | -0.444 ± 0.834 | — |
| theta[43] | -0.441 ± 0.840 | — |
| theta[44] | -0.411 ± 0.821 | — |
| theta[45] | -0.476 ± 0.844 | — |
| theta[46] | -0.420 ± 0.770 | — |
| theta[47] | -0.464 ± 0.833 | — |
| theta[48] | -0.446 ± 0.853 | — |
| theta[49] | -0.457 ± 0.789 | — |
| theta[50] | -0.423 ± 0.875 | — |
| theta[51] | -0.408 ± 0.789 | — |
| theta[52] | -0.499 ± 0.823 | — |
| theta[53] | -0.452 ± 0.792 | — |
| theta[54] | -0.390 ± 0.793 | — |
| theta[55] | -0.474 ± 0.792 | — |
| theta[56] | -0.397 ± 0.879 | — |
| theta[57] | -0.470 ± 0.827 | — |
| theta[58] | -0.445 ± 0.825 | — |
| theta[59] | -0.425 ± 0.756 | — |
| theta[60] | -0.482 ± 0.850 | — |
| theta[61] | -0.429 ± 0.786 | — |
| theta[62] | 0.089 ± 0.887 | — |
| theta[63] | 0.047 ± 0.862 | — |
| theta[64] | 0.041 ± 0.864 | — |
| theta[65] | 0.092 ± 0.830 | — |
| theta[66] | 0.106 ± 0.856 | — |
| theta[67] | 0.121 ± 0.806 | — |
| theta[68] | 0.051 ± 0.830 | — |
| theta[69] | 0.060 ± 0.824 | — |
| theta[70] | 0.007 ± 0.863 | — |
| theta[71] | 0.129 ± 0.856 | — |
| theta[72] | 0.046 ± 0.821 | — |
| theta[73] | 0.061 ± 0.847 | — |
| theta[74] | 0.024 ± 0.833 | — |
| theta[75] | 0.067 ± 0.825 | — |
| theta[76] | 0.109 ± 0.851 | — |
| theta[77] | -1.41 ± 0.823 | — |
| theta[78] | -1.45 ± 0.812 | — |
| theta[79] | -1.38 ± 0.787 | — |
| theta[80] | -1.49 ± 0.793 | — |
| theta[81] | -1.40 ± 0.813 | — |
| theta[82] | -1.40 ± 0.821 | — |
| theta[83] | -1.38 ± 0.787 | — |
| theta[84] | -1.45 ± 0.755 | — |
| theta[85] | -1.45 ± 0.803 | — |
| theta[86] | -1.40 ± 0.763 | — |
| theta[87] | -0.895 ± 0.797 | — |
| theta[88] | -0.897 ± 0.811 | — |
| theta[89] | -1.00 ± 0.777 | — |
| theta[90] | -0.960 ± 0.808 | — |
| theta[91] | -0.926 ± 0.830 | — |
| theta[92] | -0.954 ± 0.818 | — |
| theta[93] | -0.874 ± 0.806 | — |
| theta[94] | -0.954 ± 0.825 | — |
| theta[95] | -0.989 ± 0.826 | — |
| theta[96] | -0.917 ± 0.793 | — |
| theta[97] | -0.825 ± 0.846 | — |
| theta[98] | -0.860 ± 0.795 | — |
| theta[99] | -0.966 ± 0.828 | — |
| theta[100] | -0.988 ± 0.818 | — |
| theta[101] | -0.845 ± 0.840 | — |
| theta[102] | -0.909 ± 0.788 | — |
| theta[103] | -0.918 ± 0.803 | — |
| theta[104] | -0.896 ± 0.802 | — |
| theta[105] | -0.920 ± 0.785 | — |
| theta[106] | -0.883 ± 0.825 | — |
| theta[107] | -0.938 ± 0.764 | — |
| theta[108] | -1.00 ± 0.812 | — |
| theta[109] | -0.965 ± 0.778 | — |
| theta[110] | -0.955 ± 0.829 | — |
| theta[111] | -0.928 ± 0.829 | — |
| theta[112] | -0.845 ± 0.845 | — |
| theta[113] | -0.904 ± 0.761 | — |
| theta[114] | -0.963 ± 0.828 | — |
| theta[115] | -0.945 ± 0.757 | — |
| theta[116] | -0.873 ± 0.818 | — |
| theta[117] | -0.938 ± 0.831 | — |
| theta[118] | -0.884 ± 0.789 | — |
| theta[119] | -0.810 ± 0.843 | — |
| theta[120] | -0.877 ± 0.851 | — |
| theta[121] | -0.919 ± 0.804 | — |
| theta[122] | -0.859 ± 0.800 | — |
| theta[123] | -0.971 ± 0.817 | — |
| theta[124] | -0.948 ± 0.794 | — |
| theta[125] | -0.893 ± 0.832 | — |
| theta[126] | -0.949 ± 0.810 | — |
| theta[127] | -0.900 ± 0.833 | — |
| theta[128] | -0.903 ± 0.786 | — |
| theta[129] | -0.959 ± 0.812 | — |
| theta[130] | -0.404 ± 0.858 | — |
| theta[131] | -0.448 ± 0.833 | — |
| theta[132] | -0.527 ± 0.805 | — |
| theta[133] | -0.418 ± 0.835 | — |
| theta[134] | -0.494 ± 0.830 | — |
| theta[135] | -0.401 ± 0.823 | — |
| theta[136] | -0.430 ± 0.841 | — |
| theta[137] | -0.489 ± 0.838 | — |
| theta[138] | -0.476 ± 0.825 | — |
| theta[139] | -0.460 ± 0.860 | — |
| theta[140] | -0.438 ± 0.812 | — |
| theta[141] | -0.463 ± 0.803 | — |
| theta[142] | -0.405 ± 0.835 | — |
| theta[143] | -0.464 ± 0.852 | — |
| theta[144] | -0.421 ± 0.864 | — |
| theta[145] | -0.512 ± 0.835 | — |
| theta[146] | -0.423 ± 0.834 | — |
| theta[147] | -0.449 ± 0.855 | — |
| theta[148] | -0.478 ± 0.778 | — |
| theta[149] | -0.473 ± 0.860 | — |
| theta[150] | -0.444 ± 0.854 | — |
| theta[151] | -0.463 ± 0.810 | — |
| theta[152] | -0.411 ± 0.852 | — |
| theta[153] | -0.472 ± 0.859 | — |
| theta[154] | -0.436 ± 0.834 | — |
| theta[155] | -0.448 ± 0.804 | — |
| theta[156] | -0.429 ± 0.866 | — |
| theta[157] | -0.506 ± 0.871 | — |
| theta[158] | -0.419 ± 0.828 | — |
| theta[159] | -0.490 ± 0.861 | — |
| theta[160] | -0.408 ± 0.793 | — |
| theta[161] | -0.405 ± 0.829 | — |
| theta[162] | -0.447 ± 0.865 | — |
| theta[163] | -0.454 ± 0.815 | — |
| theta[164] | -0.496 ± 0.860 | — |
| theta[165] | -0.427 ± 0.803 | — |
| theta[166] | -0.423 ± 0.813 | — |
| theta[167] | -0.452 ± 0.802 | — |
| theta[168] | -0.427 ± 0.835 | — |
| theta[169] | -0.461 ± 0.814 | — |
| theta[170] | -0.472 ± 0.802 | — |
| theta[171] | -0.416 ± 0.837 | — |
| theta[172] | -0.430 ± 0.831 | — |
| theta[173] | -0.506 ± 0.759 | — |
| theta[174] | -0.489 ± 0.832 | — |
| theta[175] | -0.433 ± 0.818 | — |
| theta[176] | -0.451 ± 0.794 | — |
| theta[177] | -0.363 ± 0.800 | — |
| theta[178] | -0.522 ± 0.818 | — |
| theta[179] | -0.435 ± 0.800 | — |
| theta[180] | -0.403 ± 0.816 | — |
| theta[181] | -0.498 ± 0.793 | — |
| theta[182] | -0.477 ± 0.796 | — |
| theta[183] | -0.406 ± 0.807 | — |
| theta[184] | -0.504 ± 0.748 | — |
| theta[185] | -0.400 ± 0.812 | — |
| theta[186] | -0.362 ± 0.811 | — |
| theta[187] | -0.471 ± 0.808 | — |
| theta[188] | -0.466 ± 0.896 | — |
| theta[189] | -0.482 ± 0.829 | — |
| theta[190] | -0.420 ± 0.846 | — |
| theta[191] | -0.448 ± 0.840 | — |
| theta[192] | -0.441 ± 0.867 | — |
| theta[193] | -0.454 ± 0.807 | — |
| theta[194] | -0.493 ± 0.829 | — |
| theta[195] | -0.474 ± 0.798 | — |
| theta[196] | -0.462 ± 0.868 | — |
| theta[197] | -0.433 ± 0.846 | — |
| theta[198] | -0.454 ± 0.818 | — |
| theta[199] | -0.371 ± 0.831 | — |
| theta[200] | -0.507 ± 0.847 | — |
| theta[201] | -0.394 ± 0.773 | — |
| theta[202] | -0.375 ± 0.845 | — |
| theta[203] | -0.420 ± 0.829 | — |
| theta[204] | -0.429 ± 0.816 | — |
| theta[205] | -0.401 ± 0.802 | — |
| theta[206] | -0.398 ± 0.790 | — |
| theta[207] | -0.411 ± 0.821 | — |
| theta[208] | -0.380 ± 0.849 | — |
| theta[209] | -0.401 ± 0.796 | — |
| theta[210] | -0.344 ± 0.870 | — |
| theta[211] | -0.848 ± 0.827 | — |
| theta[212] | -0.944 ± 0.823 | — |
| theta[213] | -0.910 ± 0.817 | — |
| theta[214] | -0.480 ± 0.790 | — |
| theta[215] | -0.479 ± 0.865 | — |
| theta[216] | -0.483 ± 0.823 | — |
| theta[217] | -0.452 ± 0.878 | — |
| theta[218] | -0.437 ± 0.838 | — |
| theta[219] | -0.417 ± 0.818 | — |
| theta[220] | -0.455 ± 0.785 | — |
| theta[221] | -0.458 ± 0.839 | — |
| theta[222] | -0.441 ± 0.785 | — |
| theta[223] | -0.371 ± 0.803 | — |
| theta[224] | -0.409 ± 0.873 | — |
| theta[225] | -0.456 ± 0.818 | — |
| theta[226] | -0.412 ± 0.803 | — |
| theta[227] | -0.412 ± 0.817 | — |
| theta[228] | -0.451 ± 0.844 | — |
| theta[229] | -0.415 ± 0.844 | — |
| theta[230] | -0.458 ± 0.857 | — |
| theta[231] | -0.471 ± 0.834 | — |
| theta[232] | -0.447 ± 0.854 | — |
| theta[233] | -0.482 ± 0.828 | — |
| theta[234] | -0.451 ± 0.800 | — |
| theta[235] | -0.458 ± 0.832 | — |
| theta[236] | -0.440 ± 0.840 | — |
| theta[237] | -0.462 ± 0.810 | — |
| theta[238] | -0.420 ± 0.810 | — |
| theta[239] | -0.440 ± 0.810 | — |
| theta[240] | -0.442 ± 0.808 | — |
| theta[241] | -0.479 ± 0.854 | — |
| theta[242] | -0.408 ± 0.803 | — |
| theta[243] | -0.422 ± 0.811 | — |
| theta[244] | -0.416 ± 0.825 | — |
| theta[245] | -0.446 ± 0.854 | — |
| theta[246] | -0.423 ± 0.804 | — |
| theta[247] | -0.453 ± 0.810 | — |
| theta[248] | -0.418 ± 0.829 | — |
| theta[249] | -0.453 ± 0.785 | — |
| theta[250] | -0.430 ± 0.789 | — |
| theta[251] | -0.464 ± 0.794 | — |
| theta[252] | -0.496 ± 0.831 | — |
| theta[253] | -0.484 ± 0.770 | — |
| theta[254] | -0.399 ± 0.840 | — |
| theta[255] | -0.446 ± 0.814 | — |
| theta[256] | -0.480 ± 0.805 | — |
| theta[257] | 0.088 ± 0.805 | — |
| theta[258] | 0.086 ± 0.832 | — |
| theta[259] | 0.072 ± 0.862 | — |
| theta[260] | 0.101 ± 0.852 | — |
| theta[261] | 0.057 ± 0.866 | — |
| theta[262] | 0.058 ± 0.873 | — |
| theta[263] | 0.114 ± 0.833 | — |
| theta[264] | 0.087 ± 0.867 | — |
| theta[265] | 0.067 ± 0.842 | — |
| theta[266] | 0.083 ± 0.873 | — |
| theta[267] | 0.119 ± 0.810 | — |
| theta[268] | 0.064 ± 0.799 | — |
| theta[269] | 0.087 ± 0.849 | — |
| theta[270] | 0.017 ± 0.810 | — |
| theta[271] | 0.018 ± 0.885 | — |
| theta[272] | 0.107 ± 0.836 | — |
| theta[273] | 0.098 ± 0.794 | — |
| theta[274] | 0.123 ± 0.874 | — |
| theta[275] | 0.093 ± 0.825 | — |
| theta[276] | 0.126 ± 0.859 | — |
| theta[277] | 0.114 ± 0.846 | — |
| theta[278] | 0.081 ± 0.833 | — |
| theta[279] | 0.011 ± 0.837 | — |
| theta[280] | 0.078 ± 0.856 | — |
| theta[281] | 0.156 ± 0.843 | — |
| theta[282] | 0.119 ± 0.829 | — |
| theta[283] | 0.091 ± 0.854 | — |
| theta[284] | 0.110 ± 0.816 | — |
| theta[285] | 0.116 ± 0.902 | — |
| theta[286] | 0.060 ± 0.838 | — |
| theta[287] | 0.107 ± 0.837 | — |
| theta[288] | 0.114 ± 0.861 | — |
| theta[289] | 0.126 ± 0.800 | — |
| theta[290] | 0.166 ± 0.815 | — |
| theta[291] | 0.092 ± 0.884 | — |
| theta[292] | 0.064 ± 0.837 | — |
| theta[293] | 0.062 ± 0.847 | — |
| theta[294] | 0.041 ± 0.829 | — |
| theta[295] | 0.079 ± 0.873 | — |
| theta[296] | 0.048 ± 0.839 | — |
| theta[297] | 0.013 ± 0.807 | — |
| theta[298] | 0.082 ± 0.810 | — |
| theta[299] | 0.094 ± 0.862 | — |
| theta[300] | 0.067 ± 0.822 | — |
| theta[301] | 0.131 ± 0.824 | — |
| theta[302] | 0.088 ± 0.792 | — |
| theta[303] | 0.087 ± 0.829 | — |
| theta[304] | 0.060 ± 0.869 | — |
| theta[305] | 0.020 ± 0.923 | — |
| theta[306] | 0.116 ± 0.810 | — |
| theta[307] | 0.055 ± 0.777 | — |
| theta[308] | 0.102 ± 0.897 | — |
| theta[309] | 0.090 ± 0.848 | — |
| theta[310] | 0.171 ± 0.872 | — |
| theta[311] | 0.078 ± 0.825 | — |
| theta[312] | 0.067 ± 0.819 | — |
| theta[313] | 0.064 ± 0.848 | — |
| theta[314] | 0.096 ± 0.866 | — |
| theta[315] | 0.022 ± 0.866 | — |
| theta[316] | 0.119 ± 0.840 | — |
| theta[317] | 0.075 ± 0.820 | — |
| theta[318] | 0.093 ± 0.818 | — |
| theta[319] | 0.128 ± 0.868 | — |
| theta[320] | 0.058 ± 0.870 | — |
| theta[321] | 0.056 ± 0.885 | — |
| theta[322] | 0.094 ± 0.863 | — |
| theta[323] | 0.051 ± 0.900 | — |
| theta[324] | 0.091 ± 0.831 | — |
| theta[325] | 0.042 ± 0.854 | — |
| theta[326] | 0.105 ± 0.838 | — |
| theta[327] | 0.120 ± 0.861 | — |
| theta[328] | 0.069 ± 0.821 | — |
| theta[329] | 0.062 ± 0.830 | — |
| theta[330] | 0.126 ± 0.895 | — |
| theta[331] | 0.064 ± 0.828 | — |
| theta[332] | 0.130 ± 0.842 | — |
| theta[333] | 0.065 ± 0.844 | — |
| theta[334] | 0.100 ± 0.851 | — |
| theta[335] | 0.130 ± 0.804 | — |
| theta[336] | 0.062 ± 0.902 | — |
| theta[337] | -0.955 ± 0.796 | — |
| theta[338] | -0.907 ± 0.817 | — |
| theta[339] | -0.922 ± 0.833 | — |
| theta[340] | -0.942 ± 0.846 | — |
| theta[341] | -0.967 ± 0.795 | — |
| theta[342] | -0.977 ± 0.834 | — |
| theta[343] | -0.917 ± 0.810 | — |
| theta[344] | -0.924 ± 0.831 | — |
| theta[345] | -0.953 ± 0.822 | — |
| theta[346] | -0.845 ± 0.796 | — |
| theta[347] | -0.981 ± 0.819 | — |
| theta[348] | -0.937 ± 0.777 | — |
| theta[349] | -0.940 ± 0.778 | — |
| theta[350] | -0.946 ± 0.823 | — |
| theta[351] | -0.920 ± 0.797 | — |
| theta[352] | -0.958 ± 0.766 | — |
| theta[353] | -0.429 ± 0.869 | — |
| theta[354] | -0.403 ± 0.843 | — |
| theta[355] | -0.440 ± 0.852 | — |
| theta[356] | -0.414 ± 0.795 | — |
| theta[357] | -0.398 ± 0.817 | — |
| theta[358] | -0.447 ± 0.832 | — |
| theta[359] | -0.434 ± 0.799 | — |
| theta[360] | -0.414 ± 0.858 | — |
| theta[361] | -0.459 ± 0.846 | — |
| theta[362] | -0.373 ± 0.851 | — |
| theta[363] | -0.435 ± 0.819 | — |
| theta[364] | -0.416 ± 0.817 | — |
| theta[365] | -0.443 ± 0.834 | — |
| theta[366] | -0.388 ± 0.841 | — |
| theta[367] | -0.457 ± 0.805 | — |
| theta[368] | -0.450 ± 0.835 | — |
| theta[369] | -0.466 ± 0.796 | — |
| theta[370] | -0.451 ± 0.860 | — |
| theta[371] | -0.449 ± 0.778 | — |
| theta[372] | -0.487 ± 0.794 | — |
| theta[373] | -0.435 ± 0.826 | — |
| theta[374] | -0.386 ± 0.860 | — |
| theta[375] | -0.412 ± 0.804 | — |
| theta[376] | -0.364 ± 0.816 | — |
| theta[377] | -0.499 ± 0.803 | — |
| theta[378] | -0.438 ± 0.796 | — |
| theta[379] | -0.405 ± 0.818 | — |
| theta[380] | -0.394 ± 0.825 | — |
| theta[381] | -0.494 ± 0.794 | — |
| theta[382] | -0.461 ± 0.832 | — |
| theta[383] | -0.381 ± 0.843 | — |
| theta[384] | -0.466 ± 0.769 | — |
| theta[385] | -0.469 ± 0.774 | — |
| theta[386] | -0.422 ± 0.800 | — |
| theta[387] | -0.457 ± 0.811 | — |
| theta[388] | -0.478 ± 0.832 | — |
| theta[389] | -0.439 ± 0.782 | — |
| theta[390] | -0.481 ± 0.836 | — |
| theta[391] | -0.430 ± 0.790 | — |
| theta[392] | -0.334 ± 0.807 | — |
| theta[393] | -0.484 ± 0.793 | — |
| theta[394] | -0.412 ± 0.841 | — |
| theta[395] | -0.413 ± 0.807 | — |
| theta[396] | -0.461 ± 0.828 | — |
| theta[397] | -0.452 ± 0.836 | — |
| theta[398] | -0.526 ± 0.799 | — |
| theta[399] | -0.490 ± 0.799 | — |
| theta[400] | -0.369 ± 0.853 | — |
| theta[401] | -0.494 ± 0.806 | — |
| theta[402] | -0.389 ± 0.793 | — |
| theta[403] | -0.478 ± 0.806 | — |
| theta[404] | -0.443 ± 0.807 | — |
| theta[405] | -0.379 ± 0.792 | — |
| theta[406] | -0.426 ± 0.838 | — |
| theta[407] | -0.385 ± 0.805 | — |
| theta[408] | -0.397 ± 0.788 | — |
| theta[409] | -0.400 ± 0.802 | — |
| theta[410] | -0.395 ± 0.831 | — |
| theta[411] | -0.396 ± 0.799 | — |
| theta[412] | -0.469 ± 0.852 | — |
| theta[413] | -0.435 ± 0.813 | — |
| theta[414] | -0.509 ± 0.782 | — |
| theta[415] | -0.495 ± 0.830 | — |
| theta[416] | -0.442 ± 0.858 | — |
| theta[417] | -0.491 ± 0.828 | — |
| theta[418] | -0.431 ± 0.799 | — |
| theta[419] | -0.400 ± 0.831 | — |
| theta[420] | -0.390 ± 0.792 | — |
| theta[421] | -0.460 ± 0.813 | — |
| theta[422] | -0.400 ± 0.811 | — |
| theta[423] | -0.460 ± 0.834 | — |
| theta[424] | -0.442 ± 0.840 | — |
| theta[425] | -0.461 ± 0.849 | — |
| theta[426] | -0.472 ± 0.818 | — |
| theta[427] | -0.447 ± 0.834 | — |
| theta[428] | -0.378 ± 0.800 | — |
| theta[429] | -0.445 ± 0.799 | — |
| theta[430] | 0.024 ± 0.846 | — |
| theta[431] | 0.088 ± 0.826 | — |
| theta[432] | 0.109 ± 0.841 | — |
| theta[433] | 0.063 ± 0.834 | — |
| theta[434] | 0.037 ± 0.824 | — |
| theta[435] | 0.119 ± 0.851 | — |
| theta[436] | 0.097 ± 0.901 | — |
| theta[437] | 0.021 ± 0.831 | — |
| theta[438] | 0.052 ± 0.848 | — |
| theta[439] | 0.057 ± 0.845 | — |
| theta[440] | 0.153 ± 0.893 | — |
| theta[441] | 0.140 ± 0.861 | — |
| theta[442] | 0.143 ± 0.821 | — |
| theta[443] | 0.051 ± 0.806 | — |
| theta[444] | 0.087 ± 0.843 | — |
| theta[445] | 0.085 ± 0.850 | — |
| theta[446] | 0.060 ± 0.804 | — |
| theta[447] | 0.091 ± 0.841 | — |
| theta[448] | 0.084 ± 0.835 | — |
| theta[449] | 0.067 ± 0.815 | — |
| theta[450] | 0.066 ± 0.861 | — |
| theta[451] | 0.092 ± 0.867 | — |
| theta[452] | 0.136 ± 0.848 | — |
| theta[453] | 0.143 ± 0.887 | — |
| theta[454] | 0.073 ± 0.821 | — |
| theta[455] | 0.131 ± 0.809 | — |
| theta[456] | 0.097 ± 0.841 | — |
| theta[457] | 0.076 ± 0.804 | — |
| theta[458] | 0.113 ± 0.839 | — |
| theta[459] | -0.002 ± 0.850 | — |
| theta[460] | 0.122 ± 0.837 | — |
| theta[461] | 0.131 ± 0.877 | — |
| theta[462] | 0.076 ± 0.890 | — |
| theta[463] | 0.017 ± 0.817 | — |
| theta[464] | 0.096 ± 0.837 | — |
| theta[465] | 0.064 ± 0.846 | — |
| theta[466] | 0.044 ± 0.824 | — |
| theta[467] | 0.067 ± 0.853 | — |
| theta[468] | 0.096 ± 0.805 | — |
| theta[469] | 0.102 ± 0.814 | — |
| theta[470] | 0.082 ± 0.899 | — |
| theta[471] | 0.093 ± 0.859 | — |
| theta[472] | 0.083 ± 0.808 | — |
| theta[473] | 0.142 ± 0.811 | — |
| theta[474] | 0.069 ± 0.788 | — |
| theta[475] | 0.010 ± 0.829 | — |
| theta[476] | 0.077 ± 0.810 | — |
| theta[477] | 0.053 ± 0.891 | — |
| theta[478] | 0.059 ± 0.840 | — |
| theta[479] | 0.092 ± 0.848 | — |
| theta[480] | 0.063 ± 0.823 | — |
| theta[481] | 0.063 ± 0.867 | — |
| theta[482] | 0.051 ± 0.837 | — |
| theta[483] | 0.070 ± 0.849 | — |
| theta[484] | 0.077 ± 0.826 | — |
| theta[485] | 0.044 ± 0.864 | — |
| theta[486] | 0.059 ± 0.795 | — |
| theta[487] | 0.050 ± 0.797 | — |
| theta[488] | 0.052 ± 0.828 | — |
| theta[489] | 0.074 ± 0.854 | — |
| theta[490] | 0.097 ± 0.835 | — |
| theta[491] | 0.059 ± 0.848 | — |
| theta[492] | 0.121 ± 0.806 | — |
| theta[493] | 0.099 ± 0.845 | — |
| theta[494] | 0.047 ± 0.839 | — |
| theta[495] | 0.027 ± 0.859 | — |
| theta[496] | 0.073 ± 0.850 | — |
| theta[497] | 0.097 ± 0.844 | — |
| theta[498] | 0.148 ± 0.828 | — |
| theta[499] | 0.123 ± 0.845 | — |
| theta[500] | 0.126 ± 0.823 | — |
| theta[501] | 0.131 ± 0.880 | — |
| theta[502] | 0.156 ± 0.875 | — |
| theta[503] | 0.065 ± 0.850 | — |
| theta[504] | 0.118 ± 0.861 | — |
| theta[505] | 0.047 ± 0.818 | — |
| theta[506] | 0.075 ± 0.848 | — |
| theta[507] | 0.176 ± 0.805 | — |
| theta[508] | 0.165 ± 0.915 | — |
| theta[509] | 0.167 ± 0.833 | — |
| theta[510] | 0.040 ± 0.859 | — |
| theta[511] | 0.020 ± 0.867 | — |
| theta[512] | 0.122 ± 0.832 | — |
| theta[513] | 0.080 ± 0.827 | — |
| theta[514] | 0.081 ± 0.823 | — |
| theta[515] | 0.058 ± 0.845 | — |
| theta[516] | 0.104 ± 0.835 | — |
| theta[517] | 0.095 ± 0.831 | — |
| theta[518] | 0.174 ± 0.817 | — |
| theta[519] | 0.076 ± 0.840 | — |
| theta[520] | 0.002 ± 0.867 | — |
| theta[521] | 0.094 ± 0.840 | — |
| theta[522] | 0.091 ± 0.850 | — |
| theta[523] | 0.040 ± 0.863 | — |
| theta[524] | 0.093 ± 0.808 | — |
| theta[525] | 0.096 ± 0.828 | — |
| theta[526] | 0.124 ± 0.871 | — |
| theta[527] | 0.079 ± 0.826 | — |
| theta[528] | 0.076 ± 0.872 | — |
| theta[529] | 0.066 ± 0.852 | — |
| theta[530] | 0.115 ± 0.870 | — |
| theta[531] | 0.116 ± 0.891 | — |
| theta[532] | 0.191 ± 0.842 | — |
| theta[533] | 0.067 ± 0.838 | — |
| theta[534] | 0.058 ± 0.847 | — |
| theta[535] | 0.077 ± 0.842 | — |
| theta[536] | 0.103 ± 0.819 | — |
| theta[537] | 0.119 ± 0.877 | — |
| theta[538] | 0.034 ± 0.817 | — |
| theta[539] | 0.134 ± 0.846 | — |
| theta[540] | 0.065 ± 0.835 | — |
| theta[541] | 0.077 ± 0.772 | — |
| theta[542] | 0.046 ± 0.837 | — |
| theta[543] | 0.056 ± 0.847 | — |
| theta[544] | 0.105 ± 0.868 | — |
| theta[545] | 0.026 ± 0.869 | — |
| theta[546] | 0.084 ± 0.862 | — |
| theta[547] | 0.097 ± 0.852 | — |
| theta[548] | 0.042 ± 0.851 | — |
| theta[549] | 0.154 ± 0.820 | — |
| theta[550] | 0.104 ± 0.862 | — |
| theta[551] | 0.076 ± 0.868 | — |
| theta[552] | 0.104 ± 0.896 | — |
| theta[553] | 0.050 ± 0.788 | — |
| theta[554] | 0.100 ± 0.876 | — |
| theta[555] | 0.080 ± 0.829 | — |
| theta[556] | 0.089 ± 0.817 | — |
| theta[557] | 0.058 ± 0.829 | — |
| theta[558] | 0.104 ± 0.849 | — |
| theta[559] | 0.090 ± 0.890 | — |
| theta[560] | 0.087 ± 0.819 | — |
| theta[561] | 0.084 ± 0.837 | — |
| theta[562] | 0.071 ± 0.859 | — |
| theta[563] | 0.105 ± 0.850 | — |
| theta[564] | 0.077 ± 0.837 | — |
| theta[565] | 0.031 ± 0.870 | — |
| theta[566] | 0.118 ± 0.858 | — |
| theta[567] | 0.053 ± 0.846 | — |
| theta[568] | 0.102 ± 0.839 | — |
| theta[569] | 0.051 ± 0.807 | — |
| theta[570] | 0.138 ± 0.835 | — |
| theta[571] | 0.029 ± 0.839 | — |
| theta[572] | 0.061 ± 0.852 | — |
| theta[573] | 0.128 ± 0.819 | — |
| theta[574] | 0.031 ± 0.827 | — |
| theta[575] | 0.060 ± 0.868 | — |
| theta[576] | 0.125 ± 0.790 | — |
| theta[577] | 0.074 ± 0.851 | — |
| theta[578] | 0.145 ± 0.811 | — |
| theta[579] | 0.090 ± 0.813 | — |
| theta[580] | 0.056 ± 0.790 | — |
| theta[581] | 0.037 ± 0.857 | — |
| theta[582] | 0.023 ± 0.818 | — |
| theta[583] | 0.113 ± 0.789 | — |
| theta[584] | 0.058 ± 0.842 | — |
| theta[585] | 0.051 ± 0.838 | — |
| theta[586] | 0.127 ± 0.844 | — |
| theta[587] | 0.042 ± 0.807 | — |
| theta[588] | 0.032 ± 0.856 | — |
| theta[589] | 0.146 ± 0.846 | — |
| theta[590] | 0.110 ± 0.850 | — |
| theta[591] | 0.088 ± 0.877 | — |
| theta[592] | 0.123 ± 0.817 | — |
| theta[593] | 0.091 ± 0.868 | — |
| theta[594] | 0.076 ± 0.788 | — |
| theta[595] | 0.001 ± 0.834 | — |
| theta[596] | -0.009 ± 0.806 | — |
| theta[597] | 0.139 ± 0.848 | — |
| theta[598] | 0.047 ± 0.817 | — |
| theta[599] | 0.027 ± 0.875 | — |
| theta[600] | 0.147 ± 0.852 | — |
| theta[601] | 0.044 ± 0.846 | — |
| theta[602] | 0.105 ± 0.800 | — |
| theta[603] | -0.413 ± 0.834 | — |
| theta[604] | -0.373 ± 0.792 | — |
| theta[605] | -0.501 ± 0.800 | — |
| theta[606] | -0.483 ± 0.812 | — |
| theta[607] | -0.461 ± 0.827 | — |
| theta[608] | -0.392 ± 0.851 | — |
| theta[609] | -0.391 ± 0.776 | — |
| theta[610] | -0.510 ± 0.862 | — |
| theta[611] | -0.455 ± 0.838 | — |
| theta[612] | -0.418 ± 0.813 | — |
| theta[613] | -0.366 ± 0.808 | — |
| theta[614] | 0.141 ± 0.848 | — |
| theta[615] | 0.072 ± 0.847 | — |
| theta[616] | 0.075 ± 0.809 | — |
| theta[617] | 0.103 ± 0.814 | — |
| theta[618] | 0.087 ± 0.866 | — |
| theta[619] | 0.073 ± 0.881 | — |
| theta[620] | 0.093 ± 0.843 | — |
| theta[621] | 0.111 ± 0.849 | — |
| theta[622] | 0.159 ± 0.885 | — |
| theta[623] | 0.074 ± 0.846 | — |
| theta[624] | 0.015 ± 0.784 | — |
| theta[625] | 0.079 ± 0.844 | — |
| theta[626] | 0.160 ± 0.825 | — |
| theta[627] | 0.156 ± 0.825 | — |
| theta[628] | 0.056 ± 0.824 | — |
| theta[629] | 0.042 ± 0.813 | — |
| theta[630] | 0.067 ± 0.822 | — |
| theta[631] | 0.108 ± 0.856 | — |
| theta[632] | 0.030 ± 0.772 | — |
| theta[633] | 0.036 ± 0.830 | — |
| theta[634] | 0.043 ± 0.846 | — |
| theta[635] | 0.088 ± 0.871 | — |
| theta[636] | 0.080 ± 0.834 | — |
| theta[637] | 0.060 ± 0.844 | — |
| theta[638] | 0.132 ± 0.775 | — |
| theta[639] | 0.065 ± 0.854 | — |
| theta[640] | 0.096 ± 0.831 | — |
| theta[641] | 0.123 ± 0.873 | — |
| theta[642] | 0.056 ± 0.799 | — |
| theta[643] | 0.089 ± 0.885 | — |
| theta[644] | 0.022 ± 0.785 | — |
| theta[645] | 0.086 ± 0.866 | — |
| theta[646] | 0.035 ± 0.832 | — |
| theta[647] | 0.107 ± 0.805 | — |
| theta[648] | 0.018 ± 0.884 | — |
| theta[649] | 0.034 ± 0.773 | — |
| theta[650] | 0.065 ± 0.845 | — |
| theta[651] | 0.096 ± 0.862 | — |
| theta[652] | 0.110 ± 0.824 | — |
| theta[653] | 0.079 ± 0.835 | — |
| theta[654] | 0.099 ± 0.826 | — |
| theta[655] | 0.097 ± 0.814 | — |
| theta[656] | 0.049 ± 0.842 | — |
| theta[657] | 0.076 ± 0.842 | — |
| theta[658] | 0.092 ± 0.846 | — |
| theta[659] | 0.090 ± 0.814 | — |
| theta[660] | 0.143 ± 0.832 | — |
| theta[661] | 0.052 ± 0.874 | — |
| theta[662] | 0.143 ± 0.847 | — |
| theta[663] | 0.136 ± 0.850 | — |
| theta[664] | 0.088 ± 0.809 | — |
| theta[665] | 0.080 ± 0.818 | — |
| theta[666] | 0.053 ± 0.837 | — |
| theta[667] | 0.075 ± 0.858 | — |
| theta[668] | 0.103 ± 0.859 | — |
| theta[669] | 0.098 ± 0.787 | — |
| theta[670] | 0.091 ± 0.876 | — |
| theta[671] | 0.153 ± 0.856 | — |
| theta[672] | 0.085 ± 0.827 | — |
| theta[673] | 0.144 ± 0.849 | — |
| theta[674] | 0.072 ± 0.894 | — |
| theta[675] | 0.056 ± 0.848 | — |
| theta[676] | 0.036 ± 0.813 | — |
| theta[677] | 0.077 ± 0.833 | — |
| theta[678] | 0.167 ± 0.828 | — |
| theta[679] | 0.114 ± 0.854 | — |
| theta[680] | 0.082 ± 0.805 | — |
| theta[681] | 0.040 ± 0.847 | — |
| theta[682] | 0.127 ± 0.842 | — |
| theta[683] | 0.075 ± 0.838 | — |
| theta[684] | 0.106 ± 0.830 | — |
| theta[685] | 0.139 ± 0.879 | — |
| theta[686] | 0.104 ± 0.833 | — |
| theta[687] | 0.121 ± 0.874 | — |
| theta[688] | 0.142 ± 0.778 | — |
| theta[689] | 0.094 ± 0.833 | — |
| theta[690] | 0.047 ± 0.818 | — |
| theta[691] | 0.085 ± 0.832 | — |
| theta[692] | 0.043 ± 0.818 | — |
| theta[693] | 0.047 ± 0.848 | — |
| theta[694] | 0.070 ± 0.823 | — |
| theta[695] | 0.069 ± 0.840 | — |
| theta[696] | 0.093 ± 0.854 | — |
| theta[697] | 0.144 ± 0.788 | — |
| theta[698] | 0.116 ± 0.765 | — |
| theta[699] | 0.086 ± 0.816 | — |
| theta[700] | 9.26e-4 ± 0.837 | — |
| theta[701] | 0.041 ± 0.851 | — |
| theta[702] | 0.136 ± 0.843 | — |
| theta[703] | 0.642 ± 0.911 | — |
| theta[704] | 0.596 ± 0.888 | — |
| theta[705] | 0.660 ± 0.879 | — |
| theta[706] | 0.659 ± 0.871 | — |
| theta[707] | 0.639 ± 0.890 | — |
| theta[708] | 0.660 ± 0.882 | — |
| theta[709] | 0.647 ± 0.868 | — |
| theta[710] | 0.618 ± 0.905 | — |
| theta[711] | 0.601 ± 0.860 | — |
| theta[712] | 0.549 ± 0.815 | — |
| theta[713] | 0.596 ± 0.834 | — |
| theta[714] | 0.632 ± 0.811 | — |
| theta[715] | 0.693 ± 0.891 | — |
| theta[716] | 0.624 ± 0.851 | — |
| theta[717] | 0.730 ± 0.852 | — |
| theta[718] | 0.588 ± 0.819 | — |
| theta[719] | 0.621 ± 0.908 | — |
| theta[720] | 0.609 ± 0.874 | — |
| theta[721] | 0.645 ± 0.840 | — |
| theta[722] | 0.613 ± 0.856 | — |
| theta[723] | 0.583 ± 0.864 | — |
| theta[724] | 0.636 ± 0.879 | — |
| theta[725] | 0.637 ± 0.872 | — |
| theta[726] | 0.630 ± 0.854 | — |
| theta[727] | 0.612 ± 0.872 | — |
| theta[728] | 0.567 ± 0.826 | — |
| theta[729] | 0.650 ± 0.842 | — |
| theta[730] | 0.620 ± 0.928 | — |
| theta[731] | 0.672 ± 0.820 | — |
| theta[732] | 0.635 ± 0.812 | — |
| theta[733] | 0.519 ± 0.871 | — |
| theta[734] | 0.621 ± 0.854 | — |
| theta[735] | 0.703 ± 0.834 | — |
| theta[736] | 0.590 ± 0.898 | — |
| theta[737] | 0.618 ± 0.869 | — |
| theta[738] | 0.710 ± 0.880 | — |
| theta[739] | 0.655 ± 0.901 | — |
| theta[740] | 0.598 ± 0.838 | — |
| theta[741] | 0.662 ± 0.888 | — |
| theta[742] | 0.518 ± 0.876 | — |
| theta[743] | 0.596 ± 0.871 | — |
| theta[744] | 0.716 ± 0.907 | — |
| theta[745] | 0.644 ± 0.882 | — |
| theta[746] | 0.629 ± 0.877 | — |
| theta[747] | 0.634 ± 0.841 | — |
| theta[748] | 0.647 ± 0.878 | — |
| theta[749] | 0.638 ± 0.876 | — |
| theta[750] | 0.714 ± 0.873 | — |
| theta[751] | 0.581 ± 0.854 | — |
| theta[752] | 0.601 ± 0.812 | — |
| theta[753] | 0.623 ± 0.882 | — |
| theta[754] | 0.590 ± 0.869 | — |
| theta[755] | 0.613 ± 0.836 | — |
| theta[756] | 0.728 ± 0.822 | — |
| theta[757] | 0.639 ± 0.873 | — |
| theta[758] | 0.598 ± 0.894 | — |
| theta[759] | 0.642 ± 0.857 | — |
| theta[760] | 0.686 ± 0.843 | — |
| theta[761] | 0.602 ± 0.849 | — |
| theta[762] | 0.597 ± 0.851 | — |
| theta[763] | 0.655 ± 0.886 | — |
| theta[764] | 0.687 ± 0.870 | — |
| theta[765] | 0.606 ± 0.862 | — |
| theta[766] | 0.630 ± 0.877 | — |
| theta[767] | 0.627 ± 0.899 | — |
| theta[768] | 0.609 ± 0.836 | — |
| theta[769] | 0.649 ± 0.871 | — |
| theta[770] | 0.691 ± 0.852 | — |
| theta[771] | 0.586 ± 0.890 | — |
| theta[772] | 0.568 ± 0.824 | — |
| theta[773] | 0.625 ± 0.863 | — |
| theta[774] | 0.625 ± 0.857 | — |
| theta[775] | 0.573 ± 0.819 | — |
| theta[776] | 0.696 ± 0.869 | — |
| theta[777] | 0.627 ± 0.875 | — |
| theta[778] | 0.672 ± 0.843 | — |
| theta[779] | 0.578 ± 0.844 | — |
| theta[780] | 0.587 ± 0.871 | — |
| theta[781] | 0.608 ± 0.907 | — |
| theta[782] | 0.627 ± 0.848 | — |
| theta[783] | 0.614 ± 0.836 | — |
| theta[784] | 0.623 ± 0.879 | — |
| theta[785] | 0.610 ± 0.815 | — |
| theta[786] | 0.624 ± 0.874 | — |
| theta[787] | 0.644 ± 0.831 | — |
| theta[788] | 0.636 ± 0.850 | — |
| theta[789] | 0.638 ± 0.852 | — |
| theta[790] | 0.584 ± 0.873 | — |
| theta[791] | 0.620 ± 0.884 | — |
| theta[792] | 0.609 ± 0.853 | — |
| theta[793] | 0.708 ± 0.882 | — |
| theta[794] | 0.596 ± 0.822 | — |
| theta[795] | 0.645 ± 0.904 | — |
| theta[796] | 0.697 ± 0.884 | — |
| theta[797] | 0.601 ± 0.858 | — |
| theta[798] | 0.599 ± 0.864 | — |
| theta[799] | 0.646 ± 0.825 | — |
| theta[800] | 0.659 ± 0.846 | — |
| theta[801] | 0.615 ± 0.860 | — |
| theta[802] | 0.562 ± 0.859 | — |
| theta[803] | 0.652 ± 0.869 | — |
| theta[804] | 0.635 ± 0.898 | — |
| theta[805] | 0.607 ± 0.917 | — |
| theta[806] | 0.631 ± 0.841 | — |
| theta[807] | 0.628 ± 0.843 | — |
| theta[808] | 0.645 ± 0.850 | — |
| theta[809] | 0.581 ± 0.866 | — |
| theta[810] | 0.687 ± 0.853 | — |
| theta[811] | 0.593 ± 0.858 | — |
| theta[812] | 0.674 ± 0.873 | — |
| theta[813] | 0.635 ± 0.894 | — |
| theta[814] | 0.592 ± 0.843 | — |
| theta[815] | 0.644 ± 0.804 | — |
| theta[816] | 0.642 ± 0.868 | — |
| theta[817] | 0.614 ± 0.883 | — |
| theta[818] | 0.634 ± 0.855 | — |
| theta[819] | 0.642 ± 0.824 | — |
| theta[820] | 0.603 ± 0.850 | — |
| theta[821] | 0.594 ± 0.787 | — |
| theta[822] | 0.553 ± 0.891 | — |
| theta[823] | 0.659 ± 0.843 | — |
| theta[824] | 0.646 ± 0.835 | — |
| theta[825] | 0.650 ± 0.911 | — |
| theta[826] | 0.640 ± 0.885 | — |
| theta[827] | 0.646 ± 0.892 | — |
| theta[828] | 0.714 ± 0.861 | — |
| theta[829] | 0.682 ± 0.858 | — |
| theta[830] | 0.628 ± 0.812 | — |
| theta[831] | 0.647 ± 0.836 | — |
| theta[832] | 0.629 ± 0.865 | — |
| theta[833] | 0.676 ± 0.857 | — |
| theta[834] | 0.699 ± 0.851 | — |
| theta[835] | 0.658 ± 0.892 | — |
| theta[836] | 0.630 ± 0.865 | — |
| theta[837] | 0.609 ± 0.823 | — |
| theta[838] | 0.672 ± 0.881 | — |
| theta[839] | 0.576 ± 0.856 | — |
| theta[840] | 0.630 ± 0.882 | — |
| theta[841] | 0.638 ± 0.882 | — |
| theta[842] | 0.651 ± 0.851 | — |
| theta[843] | 0.666 ± 0.851 | — |
| theta[844] | 0.642 ± 0.881 | — |
| theta[845] | 0.598 ± 0.865 | — |
| theta[846] | 0.565 ± 0.865 | — |
| theta[847] | 0.609 ± 0.861 | — |
| theta[848] | 0.686 ± 0.889 | — |
| theta[849] | 0.655 ± 0.864 | — |
| theta[850] | 0.639 ± 0.823 | — |
| theta[851] | 0.613 ± 0.885 | — |
| theta[852] | 0.638 ± 0.868 | — |
| theta[853] | 0.622 ± 0.885 | — |
| theta[854] | 0.604 ± 0.815 | — |
| theta[855] | 0.546 ± 0.881 | — |
| theta[856] | 0.607 ± 0.888 | — |
| theta[857] | 0.635 ± 0.918 | — |
| theta[858] | 0.630 ± 0.904 | — |
| theta[859] | 0.616 ± 0.903 | — |
| theta[860] | 0.633 ± 0.840 | — |
| theta[861] | 0.599 ± 0.845 | — |
| theta[862] | 0.610 ± 0.841 | — |
| theta[863] | 0.612 ± 0.857 | — |
| theta[864] | 0.680 ± 0.890 | — |
| theta[865] | 0.630 ± 0.897 | — |
| theta[866] | 0.650 ± 0.857 | — |
| theta[867] | 0.637 ± 0.876 | — |
| theta[868] | 0.582 ± 0.851 | — |
| theta[869] | 0.602 ± 0.882 | — |
| theta[870] | 0.604 ± 0.894 | — |
| theta[871] | 0.677 ± 0.828 | — |
| theta[872] | 0.631 ± 0.849 | — |
| theta[873] | 0.701 ± 0.886 | — |
| theta[874] | 0.591 ± 0.919 | — |
| theta[875] | 0.667 ± 0.849 | — |
| theta[876] | 0.604 ± 0.832 | — |
| theta[877] | 0.675 ± 0.809 | — |
| theta[878] | 0.585 ± 0.862 | — |
| theta[879] | 0.606 ± 0.881 | — |
| theta[880] | 0.634 ± 0.857 | — |
| theta[881] | 0.630 ± 0.843 | — |
| theta[882] | 0.609 ± 0.840 | — |
| theta[883] | 0.613 ± 0.851 | — |
| theta[884] | 0.562 ± 0.828 | — |
| theta[885] | 0.673 ± 0.872 | — |
| theta[886] | 0.582 ± 0.800 | — |
| theta[887] | 0.606 ± 0.869 | — |
| theta[888] | 0.726 ± 0.859 | — |
| theta[889] | 0.642 ± 0.872 | — |
| theta[890] | 0.624 ± 0.880 | — |
| theta[891] | 0.607 ± 0.870 | — |
| theta[892] | 0.632 ± 0.903 | — |
| theta[893] | 0.619 ± 0.895 | — |
| theta[894] | 0.645 ± 0.888 | — |
| theta[895] | 0.656 ± 0.906 | — |
| theta[896] | 0.674 ± 0.866 | — |
| theta[897] | 0.616 ± 0.885 | — |
| theta[898] | 0.640 ± 0.830 | — |
| theta[899] | 0.612 ± 0.872 | — |
| theta[900] | 0.572 ± 0.835 | — |
| theta[901] | 0.611 ± 0.909 | — |
| theta[902] | 0.578 ± 0.864 | — |
| theta[903] | 0.565 ± 0.877 | — |
| theta[904] | 0.597 ± 0.861 | — |
| theta[905] | 0.597 ± 0.851 | — |
| theta[906] | 0.644 ± 0.893 | — |
| theta[907] | 0.596 ± 0.846 | — |
| theta[908] | 0.624 ± 0.885 | — |
| theta[909] | 0.634 ± 0.886 | — |
| theta[910] | 0.607 ± 0.848 | — |
| theta[911] | 0.642 ± 0.893 | — |
| theta[912] | 0.654 ± 0.815 | — |
| theta[913] | 0.648 ± 0.854 | — |
| theta[914] | 0.725 ± 0.832 | — |
| theta[915] | 0.649 ± 0.870 | — |
| theta[916] | 0.727 ± 0.865 | — |
| theta[917] | 0.551 ± 0.900 | — |
| theta[918] | 0.649 ± 0.867 | — |
| theta[919] | 0.588 ± 0.861 | — |
| theta[920] | 0.660 ± 0.869 | — |
| theta[921] | 0.575 ± 0.875 | — |
| theta[922] | 0.635 ± 0.865 | — |
| theta[923] | 0.599 ± 0.864 | — |
| theta[924] | 0.593 ± 0.876 | — |
| theta[925] | 0.617 ± 0.834 | — |
| theta[926] | 0.668 ± 0.906 | — |
| theta[927] | 0.636 ± 0.848 | — |
| theta[928] | 0.622 ± 0.839 | — |
| theta[929] | 0.643 ± 0.882 | — |
| theta[930] | 0.702 ± 0.898 | — |
| theta[931] | 0.641 ± 0.876 | — |
| theta[932] | 0.706 ± 0.879 | — |
| theta[933] | 0.670 ± 0.858 | — |
| theta[934] | 0.653 ± 0.888 | — |
| theta[935] | 0.632 ± 0.884 | — |
| theta[936] | 0.637 ± 0.819 | — |
| theta[937] | 0.676 ± 0.860 | — |
| theta[938] | 0.607 ± 0.781 | — |
| theta[939] | 0.631 ± 0.926 | — |
| theta[940] | 0.693 ± 0.864 | — |
| theta[941] | 0.566 ± 0.858 | — |
| theta[942] | 0.628 ± 0.822 | — |
| theta[943] | 0.617 ± 0.893 | — |
| theta[944] | 0.633 ± 0.823 | — |
| theta[945] | 0.593 ± 0.873 | — |
| theta[946] | 0.604 ± 0.862 | — |
| theta[947] | 0.686 ± 0.883 | — |
| theta[948] | 0.616 ± 0.874 | — |
| theta[949] | 0.631 ± 0.850 | — |
| theta[950] | 0.561 ± 0.876 | — |
| theta[951] | 0.532 ± 0.872 | — |
| theta[952] | 0.602 ± 0.828 | — |
| theta[953] | 0.660 ± 0.867 | — |
| theta[954] | 0.597 ± 0.843 | — |
| theta[955] | 0.636 ± 0.867 | — |
| theta[956] | 0.612 ± 0.845 | — |
| theta[957] | 0.584 ± 0.882 | — |
| theta[958] | 0.649 ± 0.910 | — |
| theta[959] | 0.642 ± 0.832 | — |
| theta[960] | 0.627 ± 0.795 | — |
| theta[961] | 0.628 ± 0.831 | — |
| theta[962] | 0.571 ± 0.870 | — |
| theta[963] | 0.672 ± 0.861 | — |
| theta[964] | 0.622 ± 0.807 | — |
| theta[965] | 0.659 ± 0.834 | — |
| theta[966] | 0.598 ± 0.870 | — |
| theta[967] | 0.648 ± 0.866 | — |
| theta[968] | 0.641 ± 0.859 | — |
| theta[969] | 0.600 ± 0.838 | — |
| theta[970] | 0.629 ± 0.905 | — |
| theta[971] | 0.622 ± 0.901 | — |
| theta[972] | 0.688 ± 0.903 | — |
| theta[973] | 0.639 ± 0.892 | — |
| theta[974] | 0.650 ± 0.851 | — |
| theta[975] | 0.663 ± 0.883 | — |
| theta[976] | 0.680 ± 0.881 | — |
| theta[977] | 0.568 ± 0.861 | — |
| theta[978] | 0.646 ± 0.826 | — |
| theta[979] | 0.657 ± 0.845 | — |
| theta[980] | 0.618 ± 0.918 | — |
| theta[981] | 0.665 ± 0.896 | — |
| theta[982] | 0.595 ± 0.853 | — |
| theta[983] | 0.596 ± 0.881 | — |
| theta[984] | 0.616 ± 0.845 | — |
| theta[985] | 0.602 ± 0.849 | — |
| theta[986] | 0.662 ± 0.902 | — |
| theta[987] | 0.651 ± 0.837 | — |
| theta[988] | 0.637 ± 0.828 | — |
| theta[989] | 0.544 ± 0.898 | — |
| theta[990] | 0.631 ± 0.863 | — |
| theta[991] | 0.704 ± 0.821 | — |
| theta[992] | 0.681 ± 0.837 | — |
| theta[993] | 0.608 ± 0.902 | — |
| theta[994] | 0.605 ± 0.828 | — |
| theta[995] | 0.681 ± 0.867 | — |
| theta[996] | 0.572 ± 0.835 | — |
| theta[997] | 0.654 ± 0.875 | — |
| theta[998] | 0.631 ± 0.845 | — |
| theta[999] | 0.624 ± 0.875 | — |
| theta[1000] | 0.614 ± 0.834 | — |
| beta | 0.759 ± 0.072 | — |
| mean_alpha | -1.48 ± 0.054 | — |
| a[1] | -1.26 ± 0.110 | — |
| a[2] | 0.475 ± 0.067 | — |
| a[3] | 1.24 ± 0.071 | — |
| a[4] | 0.170 ± 0.071 | — |
| a[5] | -0.622 ± 0.083 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0688 ≤ tol 0.2659 · floors 0.0779/0.1023 |
For N = 133 observations, the data provide acceleration measurements Y[1], ..., Y[N] at corresponding time locations. The model employs two latent Gaussian processes: one for the mean of acceleration and one for the log-standard deviation. Each GP is approximated via spectral methods using Laplacian eigenfunctions, with 40 basis functions for the mean GP and 20 basis functions for the variance GP. For each GP, the data supply the basis matrix and the corresponding eigenvalue information required for the spectral approximation. Priors: The intercept for the mean has a Student-t prior with degrees of freedom 3, location -13, and scale 36. The mean GP's marginal standard deviation (sdgp_1), constrained positive, has a Student-t(degrees of freedom 3, location 0, scale 36) prior truncated to the positive reals. The mean GP's length-scale (lscale_1), constrained positive, has an inverse-gamma(shape 1.124909, rate 0.0177) prior. The latent coefficients for the mean GP (zgp_1[1] through zgp_1[40]) each have a standard normal(0, 1) prior. The intercept for the log-variance has a Student-t prior with degrees of freedom 3, location 0, and scale 10. The variance GP's marginal standard deviation (sdgp_sigma_1), constrained positive, has a Student-t(degrees of freedom 3, location 0, scale 36) prior truncated to the positive reals. The variance GP's length-scale (lscale_sigma_1), constrained positive, has an inverse-gamma(shape 1.124909, rate 0.0177) prior. The latent coefficients for the variance GP (zgp_sigma_1[1] through zgp_sigma_1[20]) each have a standard normal(0, 1) prior.
Each observed acceleration Y[n] is generated from a normal distribution whose mean is the sum of an intercept and a latent smooth function derived from a Gaussian process with an exponential-quadratic covariance kernel, and whose standard deviation is the exponential of a second latent function (an intercept plus another Gaussian process). The two Gaussian processes share the same parametric structure, each defined by a marginal standard deviation parameter, a length-scale parameter controlling smoothness, and a vector of latent coefficients that combine learned basis functions (Laplacian eigenfunctions at the observation time locations) to produce smooth, flexible mean and variance functions.
The marginal posterior distributions of all 62 parameters: Intercept (the mean intercept), sdgp_1 (the mean GP marginal standard deviation), lscale_1 (the mean GP length-scale), zgp_1[1] through zgp_1[40] (the 40 latent coefficients for the mean GP), Intercept_sigma (the log-variance intercept), sdgp_sigma_1 (the variance GP marginal standard deviation), lscale_sigma_1 (the variance GP length-scale), and zgp_sigma_1[1] through zgp_sigma_1[20] (the 20 latent coefficients for the variance GP).
answer spec
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// generated with brms 2.10.02functions {3 /* compute a latent Gaussian process4 * Args:5 * x: array of continuous predictor values6 * sdgp: marginal SD parameter7 * lscale: length-scale parameter8 * zgp: vector of independent standard normal variables9 * Returns:10 * a vector to be added to the linear predictor11 */12 vector gp(array[] vector x, real sdgp, vector lscale, vector zgp) {13 int Dls = rows(lscale);14 int N = size(x);15 matrix[N, N] cov;16 if (Dls == 1) {17 // one dimensional or isotropic GP18 cov = gp_exp_quad_cov(x, sdgp, lscale[1]);19 } else {20 // multi-dimensional non-isotropic GP21 cov = gp_exp_quad_cov(x[ : , 1], sdgp, lscale[1]);22 for (d in 2 : Dls) {23 cov = cov .* gp_exp_quad_cov(x[ : , d], 1, lscale[d]);24 }25 }26 for (n in 1 : N) {27 // deal with numerical non-positive-definiteness28 cov[n, n] += 1e-12;29 }30 return cholesky_decompose(cov) * zgp;31 }3233 /* Spectral density function of a Gaussian process34 * Args:35 * x: array of numeric values of dimension NB x D36 * sdgp: marginal SD parameter37 * lscale: vector of length-scale parameters38 * Returns:39 * numeric values of the function evaluated at 'x'40 */41 vector spd_cov_exp_quad(array[] vector x, real sdgp, vector lscale) {42 int NB = dims(x)[1];43 int D = dims(x)[2];44 int Dls = rows(lscale);45 vector[NB] out;46 if (Dls == 1) {47 // one dimensional or isotropic GP48 real constant = square(sdgp) * (sqrt(2 * pi()) * lscale[1]) ^ D;49 real neg_half_lscale2 = -0.5 * square(lscale[1]);50 for (m in 1 : NB) {51 out[m] = constant * exp(neg_half_lscale2 * dot_self(x[m]));52 }53 } else {54 // multi-dimensional non-isotropic GP55 real constant = square(sdgp) * sqrt(2 * pi()) ^ D * prod(lscale);56 vector[Dls] neg_half_lscale2 = -0.5 * square(lscale);57 for (m in 1 : NB) {58 out[m] = constant * exp(dot_product(neg_half_lscale2, square(x[m])));59 }60 }61 return out;62 }63 /* compute an approximate latent Gaussian process64 * Args:65 * X: Matrix of Laplacian eigen functions at the covariate values66 * sdgp: marginal SD parameter67 * lscale: vector of length-scale parameters68 * zgp: vector of independent standard normal variables69 * slambda: square root of the Laplacian eigen values70 * Returns:71 * a vector to be added to the linear predictor72 */73 vector gpa(matrix X, real sdgp, vector lscale, vector zgp,74 array[] vector slambda) {75 vector[cols(X)] diag_spd = sqrt(spd_cov_exp_quad(slambda, sdgp, lscale));76 return X * (diag_spd .* zgp);77 }78}79data {80 int<lower=1> N; // number of observations81 vector[N] Y; // response variable82 // data related to GPs83 // number of sub-GPs (equal to 1 unless 'by' was used)84 int<lower=1> Kgp_1;85 int<lower=1> Dgp_1; // GP dimension86 // number of basis functions of an approximate GP87 int<lower=1> NBgp_1;88 // approximate GP basis matrices89 matrix[N, NBgp_1] Xgp_1;90 // approximate GP eigenvalues91 array[NBgp_1] vector[Dgp_1] slambda_1;92 // data related to GPs93 // number of sub-GPs (equal to 1 unless 'by' was used)94 int<lower=1> Kgp_sigma_1;95 int<lower=1> Dgp_sigma_1; // GP dimension96 // number of basis functions of an approximate GP97 int<lower=1> NBgp_sigma_1;98 // approximate GP basis matrices99 matrix[N, NBgp_sigma_1] Xgp_sigma_1;100 // approximate GP eigenvalues101 array[NBgp_sigma_1] vector[Dgp_sigma_1] slambda_sigma_1;102 int prior_only; // should the likelihood be ignored?103}104transformed data {105106}107parameters {108 // temporary intercept for centered predictors109 real Intercept;110 // GP standard deviation parameters111 real<lower=0> sdgp_1;112 // GP length-scale parameters113 real<lower=0> lscale_1;114 // latent variables of the GP115 vector[NBgp_1] zgp_1;116 // temporary intercept for centered predictors117 real Intercept_sigma;118 // GP standard deviation parameters119 real<lower=0> sdgp_sigma_1;120 // GP length-scale parameters121 real<lower=0> lscale_sigma_1;122 // latent variables of the GP123 vector[NBgp_sigma_1] zgp_sigma_1;124}125transformed parameters {126 // vector versions of real parameters127 vector<lower=0>[Kgp_1] vsdgp_1;128 array[Kgp_1] vector<lower=0>[1] vlscale_1;129 vector<lower=0>[Kgp_sigma_1] vsdgp_sigma_1;130 array[Kgp_sigma_1] vector<lower=0>[1] vlscale_sigma_1;131 vsdgp_1[1] = sdgp_1;132 vlscale_1[1, 1] = lscale_1;133 vsdgp_sigma_1[1] = sdgp_sigma_1;134 vlscale_sigma_1[1, 1] = lscale_sigma_1;135}136model {137 // initialize linear predictor term138 vector[N] mu = Intercept + rep_vector(0, N)139 + gpa(Xgp_1, vsdgp_1[1], vlscale_1[1], zgp_1, slambda_1);140 // initialize linear predictor term141 vector[N] sigma = Intercept_sigma + rep_vector(0, N)142 + gpa(Xgp_sigma_1, vsdgp_sigma_1[1], vlscale_sigma_1[1],143 zgp_sigma_1, slambda_sigma_1);144 for (n in 1 : N) {145 // apply the inverse link function146 sigma[n] = exp(sigma[n]);147 }148 // priors including all constants149 target += student_t_lpdf(Intercept | 3, -13, 36);150 target += student_t_lpdf(vsdgp_1 | 3, 0, 36)151 - 1 * student_t_lccdf(0 | 3, 0, 36);152 target += normal_lpdf(zgp_1 | 0, 1);153 target += inv_gamma_lpdf(vlscale_1[1] | 1.124909, 0.0177);154 target += student_t_lpdf(Intercept_sigma | 3, 0, 10);155 target += student_t_lpdf(vsdgp_sigma_1 | 3, 0, 36)156 - 1 * student_t_lccdf(0 | 3, 0, 36);157 target += normal_lpdf(zgp_sigma_1 | 0, 1);158 target += inv_gamma_lpdf(vlscale_sigma_1[1] | 1.124909, 0.0177);159 // likelihood including all constants160 if (!prior_only) {161 target += normal_lpdf(Y | mu, sigma);162 }163}164generated quantities {165 // actual population-level intercept166 real b_Intercept = Intercept;167 // actual population-level intercept168 real b_sigma_Intercept = Intercept_sigma;169}170171//@ DATA { N: 133, Y: [133 values], Dgp_1: 1, NBgp_1: 40, Kgp_1: 1, Xgp_1: [133×40 matrix], slambda_1: [40×1 matrix], Dgp_sigma_1: 1, NBgp_sigma_1: 20, Kgp_sigma_1: 1, Xgp_sigma_1: [133×20 matrix], slambda_sigma_1: [20×1 matrix], prior_only: 0 } // values supplied at runtime172//@ PARAMS ["Intercept","sdgp_1","lscale_1","zgp_1[1]","zgp_1[2]","zgp_1[3]","zgp_1[4]","zgp_1[5]","zgp_1[6]","zgp_1[7]","zgp_1[8]","zgp_1[9]","zgp_1[10]","zgp_1[11]","zgp_1[12]","zgp_1[13]","zgp_1[14]","zgp_1[15]","zgp_1[16]","zgp_1[17]","zgp_1[18]","zgp_1[19]","zgp_1[20]","zgp_1[21]","zgp_1[22]","zgp_1[23]","zgp_1[24]","zgp_1[25]","zgp_1[26]","zgp_1[27]","zgp_1[28]","zgp_1[29]","zgp_1[30]","zgp_1[31]","zgp_1[32]","zgp_1[33]","zgp_1[34]","zgp_1[35]","zgp_1[36]","zgp_1[37]","zgp_1[38]","zgp_1[39]","zgp_1[40]","Intercept_sigma","sdgp_sigma_1","lscale_sigma_1","zgp_sigma_1[1]","zgp_sigma_1[2]","zgp_sigma_1[3]","zgp_sigma_1[4]","zgp_sigma_1[5]","zgp_sigma_1[6]","zgp_sigma_1[7]","zgp_sigma_1[8]","zgp_sigma_1[9]","zgp_sigma_1[10]","zgp_sigma_1[11]","zgp_sigma_1[12]","zgp_sigma_1[13]","zgp_sigma_1[14]","zgp_sigma_1[15]","zgp_sigma_1[16]","zgp_sigma_1[17]","zgp_sigma_1[18]","zgp_sigma_1[19]","zgp_sigma_1[20]"]173//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}174
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| Intercept | -10.68 ± 15.99 | -11.37 ± 15.86 |
| sdgp_1 | 43.62 ± 11.58 | 42.95 ± 10.30 |
| lscale_1 | 0.081 ± 0.016 | 0.079 ± 0.016 |
| zgp_1[1] | 0.019 ± 0.944 | -0.021 ± 0.953 |
| zgp_1[2] | -0.251 ± 0.858 | -0.288 ± 0.884 |
| zgp_1[3] | 0.400 ± 0.877 | 0.395 ± 0.937 |
| zgp_1[4] | 0.382 ± 0.752 | 0.382 ± 0.709 |
| zgp_1[5] | -0.783 ± 0.860 | -0.723 ± 0.729 |
| zgp_1[6] | -0.473 ± 0.841 | -0.453 ± 0.762 |
| zgp_1[7] | 0.704 ± 0.877 | 0.719 ± 0.763 |
| zgp_1[8] | 0.670 ± 0.771 | 0.662 ± 0.771 |
| zgp_1[9] | -0.453 ± 0.777 | -0.516 ± 0.769 |
| zgp_1[10] | -0.917 ± 0.793 | -0.933 ± 0.863 |
| zgp_1[11] | 0.485 ± 0.782 | 0.441 ± 0.851 |
| zgp_1[12] | 1.01 ± 0.868 | 0.993 ± 0.771 |
| zgp_1[13] | -0.570 ± 0.835 | -0.540 ± 0.809 |
| zgp_1[14] | -0.878 ± 0.829 | -0.879 ± 0.789 |
| zgp_1[15] | 0.509 ± 0.808 | 0.564 ± 0.826 |
| zgp_1[16] | 0.754 ± 0.801 | 0.791 ± 0.868 |
| zgp_1[17] | -0.366 ± 0.826 | -0.357 ± 0.772 |
| zgp_1[18] | -0.842 ± 0.875 | -0.793 ± 0.913 |
| zgp_1[19] | 0.287 ± 0.799 | 0.222 ± 0.812 |
| zgp_1[20] | 1.06 ± 0.800 | 1.05 ± 0.776 |
| zgp_1[21] | -0.098 ± 0.816 | -0.138 ± 0.739 |
| zgp_1[22] | -1.08 ± 0.829 | -1.05 ± 0.827 |
| zgp_1[23] | 0.077 ± 0.780 | 0.076 ± 0.805 |
| zgp_1[24] | 0.839 ± 0.830 | 0.831 ± 0.838 |
| zgp_1[25] | -0.016 ± 0.814 | -0.086 ± 0.824 |
| zgp_1[26] | -0.604 ± 0.875 | -0.598 ± 0.785 |
| zgp_1[27] | 0.021 ± 0.814 | 0.026 ± 0.838 |
| zgp_1[28] | 0.366 ± 0.867 | 0.385 ± 0.847 |
| zgp_1[29] | -0.017 ± 0.823 | 3.26e-4 ± 0.894 |
| zgp_1[30] | -0.325 ± 0.858 | -0.231 ± 0.864 |
| zgp_1[31] | -0.196 ± 0.846 | -0.143 ± 0.920 |
| zgp_1[32] | 0.207 ± 0.836 | 0.102 ± 0.845 |
| zgp_1[33] | 0.284 ± 0.873 | 0.318 ± 0.884 |
| zgp_1[34] | 0.036 ± 0.849 | 0.030 ± 0.845 |
| zgp_1[35] | -0.646 ± 0.925 | -0.578 ± 0.867 |
| zgp_1[36] | -0.213 ± 0.848 | -0.148 ± 0.906 |
| zgp_1[37] | 0.609 ± 0.884 | 0.629 ± 0.812 |
| zgp_1[38] | 0.288 ± 0.882 | 0.250 ± 0.883 |
| zgp_1[39] | -0.479 ± 0.905 | -0.423 ± 0.878 |
| zgp_1[40] | -0.386 ± 0.920 | -0.416 ± 0.867 |
| Intercept_sigma | 2.40 ± 0.636 | 2.43 ± 0.555 |
| sdgp_sigma_1 | 2.99 ± 1.57 | 2.84 ± 1.44 |
| lscale_sigma_1 | 0.031 ± 0.037 | 0.029 ± 0.032 |
| zgp_sigma_1[1] | 0.080 ± 1.02 | 0.056 ± 0.940 |
| zgp_sigma_1[2] | -0.513 ± 0.881 | -0.479 ± 0.946 |
| zgp_sigma_1[3] | -0.322 ± 0.949 | -0.345 ± 0.969 |
| zgp_sigma_1[4] | 0.825 ± 0.800 | 0.874 ± 0.768 |
| zgp_sigma_1[5] | 0.655 ± 0.843 | 0.630 ± 0.859 |
| zgp_sigma_1[6] | -0.880 ± 0.875 | -0.865 ± 0.802 |
| zgp_sigma_1[7] | -0.756 ± 0.843 | -0.767 ± 0.835 |
| zgp_sigma_1[8] | 0.430 ± 0.831 | 0.493 ± 0.869 |
| zgp_sigma_1[9] | 0.673 ± 0.871 | 0.753 ± 0.832 |
| zgp_sigma_1[10] | 0.064 ± 0.855 | 0.084 ± 0.891 |
| zgp_sigma_1[11] | -0.501 ± 0.801 | -0.442 ± 0.764 |
| zgp_sigma_1[12] | -0.504 ± 0.795 | -0.498 ± 0.847 |
| zgp_sigma_1[13] | 0.128 ± 0.823 | 0.185 ± 0.869 |
| zgp_sigma_1[14] | 0.573 ± 0.841 | 0.536 ± 0.865 |
| zgp_sigma_1[15] | 0.223 ± 0.817 | 0.274 ± 0.757 |
| zgp_sigma_1[16] | -0.312 ± 0.768 | -0.339 ± 0.831 |
| zgp_sigma_1[17] | -0.684 ± 0.800 | -0.640 ± 0.762 |
| zgp_sigma_1[18] | 0.064 ± 0.814 | 0.013 ± 0.750 |
| zgp_sigma_1[19] | 0.671 ± 0.571 | 0.675 ± 0.620 |
| zgp_sigma_1[20] | 0.407 ± 0.628 | 0.339 ± 0.567 |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.6570 ≤ tol 2.6900 · floors 0.7507/1.3450 |
For each of N = 46 mesquite trees, the data provide the tree's weight, two diameter measurements (diam1 and diam2), canopy height, total height, wood density, and a binary group indicator. All continuous measurements (weight, diam1, diam2, canopy height, total height, and density) are log-transformed before fitting. The model has seven regression coefficients (an intercept and six slopes), each with a flat (improper uniform) prior over the reals. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each tree's log-transformed weight is normally distributed around a linear predictor. The linear predictor is the sum of an intercept, plus a slope coefficient times each of six covariates: the log-transformed first diameter, log-transformed second diameter, log-transformed canopy height, log-transformed total height, log-transformed wood density, and the untransformed group indicator. The standard deviation of the normal distribution is sigma, common to all observations.
The marginal posterior distribution of each of the eight parameters: the intercept (reported as beta[1]), the slope on log-diameter 1 (reported as beta[2]), the slope on log-diameter 2 (reported as beta[3]), the slope on log-canopy height (reported as beta[4]), the slope on log-total height (reported as beta[5]), the slope on log-density (reported as beta[6]), the slope on group (reported as beta[7]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
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"beta[3]": {
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"beta[4]": {
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"beta[5]": {
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"beta[6]": {
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},
"beta[7]": {
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"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] weight;4 vector[N] diam1;5 vector[N] diam2;6 vector[N] canopy_height;7 vector[N] total_height;8 vector[N] density;9 vector[N] group;10}11transformed data {12 // log transformations13 vector[N] log_weight;14 vector[N] log_diam1;15 vector[N] log_diam2;16 vector[N] log_canopy_height;17 vector[N] log_total_height;18 vector[N] log_density;19 log_weight = log(weight);20 log_diam1 = log(diam1);21 log_diam2 = log(diam2);22 log_canopy_height = log(canopy_height);23 log_total_height = log(total_height);24 log_density = log(density);25}26parameters {27 vector[7] beta;28 real<lower=0> sigma;29}30model {31 log_weight ~ normal(beta[1] + beta[2] * log_diam1 + beta[3] * log_diam232 + beta[4] * log_canopy_height33 + beta[5] * log_total_height + beta[6] * log_density34 + beta[7] * group, sigma);35}3637//@ DATA { N: 46, canopy_height: [46 values], density: [46 values], diam1: [46 values], diam2: [46 values], group: [46 values], total_height: [46 values], weight: [46 values] } // values supplied at runtime38//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","sigma"]39//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}40
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0131 ≤ tol 0.0453 · floors 0.0203/0.0226 |
For each of N = 46 plants, the data provide the plant's dry weight in kilograms, two diameter measurements in meters (diam1 and diam2), canopy height in meters, and a binary group indicator (0 or 1). The model operates on log-transformed variables: log-weight is the natural logarithm of weight; log-canopy-volume is the logarithm of diam1 times diam2 times canopy-height; log-canopy-area is the logarithm of diam1 times diam2. The regression has four coefficients—an intercept, a slope on log-canopy-volume, a slope on log-canopy-area, and a slope on the group indicator—each with a flat (improper uniform) prior over the reals. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each plant's log-transformed weight is normally distributed with a mean equal to the intercept plus the slope coefficient on log-canopy-volume times the log-transformed canopy volume, plus the slope coefficient on log-canopy-area times the log-transformed canopy area, plus the slope coefficient on the group indicator times the group membership value, and a common standard deviation sigma across all plants.
The marginal posterior distributions of the five parameters: the intercept (reported as beta[1]), the slope on log-canopy-volume (reported as beta[2]), the slope on log-canopy-area (reported as beta[3]), the slope on the group indicator (reported as beta[4]), and the error standard deviation sigma.
answer spec
{
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"beta[1]": {
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"beta[2]": {
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"beta[4]": {
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"sigma": {
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}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] weight;4 vector[N] diam1;5 vector[N] diam2;6 vector[N] canopy_height;7 vector[N] group;8}9transformed data {10 vector[N] log_weight;11 vector[N] log_canopy_volume;12 vector[N] log_canopy_area;13 log_weight = log(weight);14 log_canopy_volume = log(diam1 .* diam2 .* canopy_height);15 log_canopy_area = log(diam1 .* diam2);16}17parameters {18 vector[4] beta;19 real<lower=0> sigma;20}21model {22 log_weight ~ normal(beta[1] + beta[2] * log_canopy_volume23 + beta[3] * log_canopy_area + beta[4] * group, sigma);24}2526//@ DATA { N: 46, canopy_height: [46 values], diam1: [46 values], diam2: [46 values], group: [46 values], weight: [46 values] } // values supplied at runtime27//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","sigma"]28//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}29
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0103 ≤ tol 0.0395 · floors 0.0115/0.0184 |
For each of N observations, the data provide the tree's weight and seven predictor measurements: two canopy diameters (diam1 and diam2) perpendicular to each other, canopy height, total height, wood density, and a grouping indicator. All predictors are derived by log-transforming combinations of the raw measurements: log-canopy-volume is the log of the product diam1 times diam2 times canopy_height; log-canopy-area is the log of diam1 times diam2; log-canopy-shape is the log of diam1 divided by diam2; log-total-height is the log of total_height; log-density is the log of density; and group is used directly without transformation. The model operates on log-weight, the natural logarithm of the tree weight. The regression has seven coefficients—an intercept beta[1] and six slopes beta[2] through beta[7] corresponding to the six transformed predictors (log-canopy-volume, log-canopy-area, log-canopy-shape, log-total-height, log-density, and group)—each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each tree's log-weight is normally distributed with a mean equal to a linear combination of the seven predictors and a common standard deviation sigma across all observations. The linear predictor is the intercept beta[1] plus beta[2] times log-canopy-volume, plus beta[3] times log-canopy-area, plus beta[4] times log-canopy-shape, plus beta[5] times log-total-height, plus beta[6] times log-density, plus beta[7] times the group indicator. The log-weight is thus generated from a normal distribution with this computed mean and standard deviation sigma.
The marginal posterior distributions of the eight parameters: the intercept beta[1], the six slope coefficients beta[2], beta[3], beta[4], beta[5], beta[6], and beta[7] (corresponding to log-canopy-volume, log-canopy-area, log-canopy-shape, log-total-height, log-density, and group respectively), and the error standard deviation sigma.
answer spec
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}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] weight;4 vector[N] diam1;5 vector[N] diam2;6 vector[N] canopy_height;7 vector[N] total_height;8 vector[N] density;9 vector[N] group;10}11transformed data {12 vector[N] log_weight;13 vector[N] log_canopy_volume;14 vector[N] log_canopy_area;15 vector[N] log_canopy_shape;16 vector[N] log_total_height;17 vector[N] log_density;18 log_weight = log(weight);19 log_canopy_volume = log(diam1 .* diam2 .* canopy_height);20 log_canopy_area = log(diam1 .* diam2);21 log_canopy_shape = log(diam1 ./ diam2);22 log_total_height = log(total_height);23 log_density = log(density);24}25parameters {26 vector[7] beta;27 real<lower=0> sigma;28}29model {30 log_weight ~ normal(beta[1] + beta[2] * log_canopy_volume31 + beta[3] * log_canopy_area32 + beta[4] * log_canopy_shape33 + beta[5] * log_total_height + beta[6] * log_density34 + beta[7] * group, sigma);35}3637//@ DATA { N: 46, canopy_height: [46 values], density: [46 values], diam1: [46 values], diam2: [46 values], group: [46 values], total_height: [46 values], weight: [46 values] } // values supplied at runtime38//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","sigma"]39//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}40
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0136 ≤ tol 0.0523 · floors 0.0152/0.0212 |
For each of N = 46 plants, the data provide measurements of the plant's weight (in kg), two diameter measurements diam1 and diam2, the height of the canopy, the total height of the plant, and a grouping indicator. The weight is transformed to the log scale to construct the response variable. The analysis uses six derived predictor variables computed from the raw measurements: log(diam1 times diam2 times canopy_height), log(diam1 times diam2), log(diam1 divided by diam2), log(total_height), and the grouping indicator itself (untransformed). All six regression coefficients beta[1] through beta[6] have flat (improper uniform) priors over the real line. The residual standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each plant's log-transformed weight is normally distributed around a linear predictor constructed from the five log-transformed geometric covariates and the grouping indicator. The linear predictor is the sum of an intercept beta[1] plus coefficients beta[2] through beta[6] multiplied by the corresponding derived predictors: beta[2] times log(diam1 times diam2 times canopy_height) plus beta[3] times log(diam1 times diam2) plus beta[4] times log(diam1 divided by diam2) plus beta[5] times log(total_height) plus beta[6] times the grouping indicator. The standard deviation of the normal distribution for each observation is the common parameter sigma across all plants.
The marginal posterior distributions of each of the seven parameters: beta[1] (the intercept), beta[2] (the coefficient for log canopy volume), beta[3] (the coefficient for log canopy area), beta[4] (the coefficient for log canopy shape), beta[5] (the coefficient for log total height), beta[6] (the coefficient for the grouping indicator), and sigma (the residual standard deviation).
answer spec
{
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}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] weight;4 vector[N] diam1;5 vector[N] diam2;6 vector[N] canopy_height;7 vector[N] total_height;8 vector[N] group;9}10transformed data {11 vector[N] log_weight;12 vector[N] log_canopy_volume;13 vector[N] log_canopy_area;14 vector[N] log_canopy_shape;15 vector[N] log_total_height;16 log_weight = log(weight);17 log_canopy_volume = log(diam1 .* diam2 .* canopy_height);18 log_canopy_area = log(diam1 .* diam2);19 log_canopy_shape = log(diam1 ./ diam2);20 log_total_height = log(total_height);21}22parameters {23 vector[6] beta;24 real<lower=0> sigma;25}26model {27 log_weight ~ normal(beta[1] + beta[2] * log_canopy_volume28 + beta[3] * log_canopy_area29 + beta[4] * log_canopy_shape30 + beta[5] * log_total_height + beta[6] * group, sigma);31}3233//@ DATA { N: 46, canopy_height: [46 values], diam1: [46 values], diam2: [46 values], group: [46 values], total_height: [46 values], weight: [46 values] } // values supplied at runtime34//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","sigma"]35//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}36
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0137 ≤ tol 0.0554 · floors 0.0215/0.0277 |
For each of N mesquite trees, the data provide the tree's weight (positive real value), two diameter measurements (diam1 and diam2, both positive reals), and canopy height (positive real). The model operates on log-transformed variables: the natural logarithm of weight as the response, and the natural logarithm of canopy volume (computed as the product diam1 * diam2 * canopy_height) as the predictor. The two regression coefficients beta[1] and beta[2] each have a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each tree's log-weight is normally distributed with a mean equal to the intercept plus the slope times the tree's log-canopy-volume, and a common standard deviation sigma across all trees.
The marginal posterior distribution of each of the three parameters: the intercept (reported as beta[1]), the slope on log-canopy-volume (reported as beta[2]), and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
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"domain": "real"
},
"beta[2]": {
"kind": "dist",
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"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] weight;4 vector[N] diam1;5 vector[N] diam2;6 vector[N] canopy_height;7}8transformed data {9 vector[N] log_weight;10 vector[N] log_canopy_volume;11 log_weight = log(weight);12 log_canopy_volume = log(diam1 .* diam2 .* canopy_height);13}14parameters {15 vector[2] beta;16 real<lower=0> sigma;17}18model {19 log_weight ~ normal(beta[1] + beta[2] * log_canopy_volume, sigma);20}2122//@ DATA { N: 46, canopy_height: [46 values], diam1: [46 values], diam2: [46 values], weight: [46 values] } // values supplied at runtime23//@ PARAMS ["beta[1]","beta[2]","sigma"]24//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}25
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0030 ≤ tol 0.0123 · floors 0.0037/0.0050 |
For each of N mesquite trees the data provide measurements of the tree's weight and six predictor variables: two diameter measurements (diam1 and diam2), canopy height, total height, wood density, and a group indicator. The regression has seven coefficients: an intercept and six slopes (one for each predictor). The intercept has a flat improper uniform prior over the real line. Each of the six predictor slopes has a flat improper uniform prior over the real line. The error standard deviation sigma, constrained to be positive, has a flat improper uniform prior over the positive reals.
Each tree's weight is Normal-distributed with a mean equal to the intercept plus the sum of six terms, each being a slope coefficient times the corresponding predictor (diam1, diam2, canopy_height, total_height, density, and group), and a common standard deviation sigma across all observations.
The marginal posterior distribution of each of the eight parameters: the intercept (reported as beta[1]), the six predictor slopes (reported as beta[2] through beta[7] for diam1, diam2, canopy_height, total_height, density, and group respectively), and the error standard deviation sigma.
answer spec
{
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"domain": "real"
},
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}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] weight;4 vector[N] diam1;5 vector[N] diam2;6 vector[N] canopy_height;7 vector[N] total_height;8 vector[N] density;9 vector[N] group;10}11parameters {12 vector[7] beta;13 real<lower=0> sigma;14}15model {16 weight ~ normal(beta[1] + beta[2] * diam1 + beta[3] * diam217 + beta[4] * canopy_height + beta[5] * total_height18 + beta[6] * density + beta[7] * group, sigma);19}2021//@ DATA { N: 46, canopy_height: [46 values], density: [46 values], diam1: [46 values], diam2: [46 values], group: [46 values], total_height: [46 values], weight: [46 values] } // values supplied at runtime22//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","sigma"]23//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}24
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=7.6474 ≤ tol 29.6690 · floors 9.2119/14.1346 |
For N = 1330 respondents, the data provide party identification score (partyid7), measured on a 7-point ordinal scale; self-reported ideology rating (real_ideo); race coding (race_adj); education level (educ1); gender (gender); income level (income); and age group (age_discrete, coded 1 = under 30, 2 = 30 to 44, 3 = 45 to 64, 4 = 65 and over). The model constructs three dummy variables for age groups: age30_44 (equals 1 if age_discrete = 2, else 0), age45_64 (equals 1 if age_discrete = 3, else 0), and age65up (equals 1 if age_discrete = 4, else 0), with the under-30 group as the reference category. The regression has nine coefficients (intercept and eight slopes), each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each respondent's party identification score is normally distributed with a mean equal to the intercept plus slopes for ideology, race, the three age-group dummy variables, education, gender, and income, each multiplied by the corresponding predictor value. The standard deviation of this normal distribution is sigma, common across all respondents.
The marginal posterior distributions of the nine parameters: the intercept (reported as beta[1]), the ideology slope (beta[2]), the race slope (beta[3]), the slope for age 30-44 (beta[4]), the slope for age 45-64 (beta[5]), the slope for age 65 and over (beta[6]), the education slope (beta[7]), the gender slope (beta[8]), the income slope (beta[9]), and the error standard deviation sigma.
answer spec
{
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"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
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}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] partyid7;4 vector[N] real_ideo;5 vector[N] race_adj;6 vector[N] educ1;7 vector[N] gender;8 vector[N] income;9 array[N] int age_discrete;10}11transformed data {12 vector[N] age30_44; // age as factor13 vector[N] age45_64;14 vector[N] age65up;1516 for (n in 1 : N) {17 age30_44[n] = age_discrete[n] == 2;18 age45_64[n] = age_discrete[n] == 3;19 age65up[n] = age_discrete[n] == 4;20 }21}22parameters {23 vector[9] beta;24 real<lower=0> sigma;25}26model {27 // vectorization28 partyid7 ~ normal(beta[1] + beta[2] * real_ideo + beta[3] * race_adj29 + beta[4] * age30_44 + beta[5] * age45_6430 + beta[6] * age65up + beta[7] * educ1 + beta[8] * gender31 + beta[9] * income, sigma);32}3334//@ DATA { N: 1330, age_discrete: [1330 values], educ1: [1330 values], gender: [1330 values], income: [1330 values], partyid7: [1330 values], race_adj: [1330 values], real_ideo: [1330 values] } // values supplied at runtime35//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","sigma"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0204 ≤ tol 0.0745 · floors 0.0372/0.0286 |
For each of N = 1184 respondents, the data provide a party identification measure on a 7-point scale (partyid7), five continuous predictors (ideological self-placement, adjusted race, education level, gender, and income), and a categorical age variable with four levels. The regression has nine coefficients: an intercept and eight slopes (for ideological self-placement, adjusted race, three binary indicators for age categories 30-44, 45-64, and 65+, education level, gender, and income), each with a flat (improper uniform) prior over the reals. The baseline age category (age under 30) is absorbed into the intercept. The error standard deviation sigma, constrained positive, has an improper uniform prior over the positive reals.
Each respondent's party identification is Normal-distributed with a mean equal to a linear combination of an intercept, the respondent's ideological self-placement, adjusted race, three binary indicators for age categories (with the baseline category under 30 absorbed into the intercept), education level, gender, and income, multiplied by their respective slope coefficients. The common standard deviation across all respondents is sigma.
The marginal posterior distribution of each of the ten parameters: the intercept (reported as beta[1]), the slope on ideological self-placement (reported as beta[2]), the slope on adjusted race (reported as beta[3]), the slope on the 30-44 age indicator (reported as beta[4]), the slope on the 45-64 age indicator (reported as beta[5]), the slope on the 65+ age indicator (reported as beta[6]), the slope on education level (reported as beta[7]), the slope on gender (reported as beta[8]), the slope on income (reported as beta[9]), and the error standard deviation sigma.
answer spec
{
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"fields": {
"beta[1]": {
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"domain": "real"
},
"beta[2]": {
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}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] partyid7;4 vector[N] real_ideo;5 vector[N] race_adj;6 vector[N] educ1;7 vector[N] gender;8 vector[N] income;9 array[N] int age_discrete;10}11transformed data {12 vector[N] age30_44; // age as factor13 vector[N] age45_64;14 vector[N] age65up;1516 for (n in 1 : N) {17 age30_44[n] = age_discrete[n] == 2;18 age45_64[n] = age_discrete[n] == 3;19 age65up[n] = age_discrete[n] == 4;20 }21}22parameters {23 vector[9] beta;24 real<lower=0> sigma;25}26model {27 // vectorization28 partyid7 ~ normal(beta[1] + beta[2] * real_ideo + beta[3] * race_adj29 + beta[4] * age30_44 + beta[5] * age45_6430 + beta[6] * age65up + beta[7] * educ1 + beta[8] * gender31 + beta[9] * income, sigma);32}3334//@ DATA { N: 1184, age_discrete: [1184 values], educ1: [1184 values], gender: [1184 values], income: [1184 values], partyid7: [1184 values], race_adj: [1184 values], real_ideo: [1184 values] } // values supplied at runtime35//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","sigma"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0131 ≤ tol 0.0395 · floors 0.0173/0.0194 |
For each of N respondents, the data provide party identification on a continuous scale (partyid7), respondent ideology (real_ideo), race adjustment (race_adj), education level (educ1), gender, income level, and a discrete age group indicator (age_discrete: 1 for under 30, 2 for ages 30-44, 3 for ages 45-64, 4 for age 65 and over). The model includes nine regression coefficients (intercept and eight slopes), each with a flat or improper uniform prior over the reals. The error standard deviation sigma, constrained positive, has an improper uniform prior over the positive reals.
Each respondent's party identification is normally distributed with a mean equal to a linear combination of an intercept, slopes on ideology, race adjustment, three age group indicators (ages 30-44, 45-64, and 65-plus), education, gender, and income, and a common standard deviation sigma across all respondents. The age group indicators are binary variables derived from the discrete age group: one for each age bracket (with the under-30 group as reference), taking value 1 if the respondent falls in that bracket and 0 otherwise.
The marginal posterior distributions of each of the ten parameters: the intercept (reported as beta[1]), the slope on ideology (reported as beta[2]), the slope on race adjustment (reported as beta[3]), the slope on the 30-44 age group indicator (reported as beta[4]), the slope on the 45-64 age group indicator (reported as beta[5]), the slope on the 65+ age group indicator (reported as beta[6]), the slope on education (reported as beta[7]), the slope on gender (reported as beta[8]), the slope on income (reported as beta[9]), and the error standard deviation sigma.
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(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] partyid7;4 vector[N] real_ideo;5 vector[N] race_adj;6 vector[N] educ1;7 vector[N] gender;8 vector[N] income;9 array[N] int age_discrete;10}11transformed data {12 vector[N] age30_44; // age as factor13 vector[N] age45_64;14 vector[N] age65up;1516 for (n in 1 : N) {17 age30_44[n] = age_discrete[n] == 2;18 age45_64[n] = age_discrete[n] == 3;19 age65up[n] = age_discrete[n] == 4;20 }21}22parameters {23 vector[9] beta;24 real<lower=0> sigma;25}26model {27 // vectorization28 partyid7 ~ normal(beta[1] + beta[2] * real_ideo + beta[3] * race_adj29 + beta[4] * age30_44 + beta[5] * age45_6430 + beta[6] * age65up + beta[7] * educ1 + beta[8] * gender31 + beta[9] * income, sigma);32}3334//@ DATA { N: 701, age_discrete: [701 values], educ1: [701 values], gender: [701 values], income: [701 values], partyid7: [701 values], race_adj: [701 values], real_ideo: [701 values] } // values supplied at runtime35//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","sigma"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0186 ≤ tol 0.0536 · floors 0.0268/0.0268 |
For each of N observations, the data provide a response variable (party identification), six continuous and categorical predictors (ideological position, race adjustment, education level, gender, and income), and a discrete age category (with values 1, 2, 3, 4 corresponding to under 30, 30-44, 45-64, and 65+). The regression includes nine coefficients: an intercept, slopes for each of the six base predictors, and slopes for three binary indicators derived from age categories (for ages 30-44, 45-64, and 65+, with ages under 30 as the implicit reference). Each of the nine coefficients has a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained to be positive, has a flat (improper uniform) prior over the positive reals.
A 9-predictor linear regression model for party identification. Each observation's party identification value is normally distributed with a mean equal to the intercept plus a weighted sum of six base predictors (ideological position, race adjustment, education level, gender, and income) and three binary age indicators (for 30-44, 45-64, and 65+, with under 30 as the reference). The standard deviation of the normal distribution is sigma, shared across all observations.
The marginal posterior distributions of the ten parameters: beta[1] (the intercept), beta[2] (the slope on ideological position), beta[3] (the slope on race adjustment), beta[4] (the slope on the age 30-44 indicator), beta[5] (the slope on the age 45-64 indicator), beta[6] (the slope on the age 65+ indicator), beta[7] (the slope on education level), beta[8] (the slope on gender), beta[9] (the slope on income), and sigma (the error standard deviation).
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(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] partyid7;4 vector[N] real_ideo;5 vector[N] race_adj;6 vector[N] educ1;7 vector[N] gender;8 vector[N] income;9 array[N] int age_discrete;10}11transformed data {12 vector[N] age30_44; // age as factor13 vector[N] age45_64;14 vector[N] age65up;1516 for (n in 1 : N) {17 age30_44[n] = age_discrete[n] == 2;18 age45_64[n] = age_discrete[n] == 3;19 age65up[n] = age_discrete[n] == 4;20 }21}22parameters {23 vector[9] beta;24 real<lower=0> sigma;25}26model {27 // vectorization28 partyid7 ~ normal(beta[1] + beta[2] * real_ideo + beta[3] * race_adj29 + beta[4] * age30_44 + beta[5] * age45_6430 + beta[6] * age65up + beta[7] * educ1 + beta[8] * gender31 + beta[9] * income, sigma);32}3334//@ DATA { N: 1226, age_discrete: [1226 values], educ1: [1226 values], gender: [1226 values], income: [1226 values], partyid7: [1226 values], race_adj: [1226 values], real_ideo: [1226 values] } // values supplied at runtime35//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","sigma"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0221 ≤ tol 0.0816 · floors 0.0408/0.0361 |
For each of N = 1113 individuals, the data provide their party identification on a 7-point scale (partyid7), ideology on a continuous scale (real_ideo), adjusted race category (race_adj), education level (educ1), gender (gender), income level (income), and age in discrete categories (age_discrete). The model has nine regression coefficients (beta[1] through beta[9]), each with a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each individual's party identification is normally distributed with a mean equal to an intercept plus eight slope terms. The slope terms are: the ideology coefficient times that individual's ideology, the race coefficient times that individual's race category, the education coefficient times that individual's education level, the gender coefficient times that individual's gender, the income coefficient times that individual's income, plus three age-related terms constructed from the discrete age variable—binary indicators for whether the individual's age is in the 30-44 range, the 45-64 range, or 65 and over, each multiplied by their respective age coefficients. All individuals share a common standard deviation sigma.
The marginal posterior distributions of each of the ten parameters: the intercept (reported as beta[1]), the ideology slope (beta[2]), the race slope (beta[3]), the age 30-44 slope (beta[4]), the age 45-64 slope (beta[5]), the age 65+ slope (beta[6]), the education slope (beta[7]), the gender slope (beta[8]), the income slope (beta[9]), and the error standard deviation sigma.
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] partyid7;4 vector[N] real_ideo;5 vector[N] race_adj;6 vector[N] educ1;7 vector[N] gender;8 vector[N] income;9 array[N] int age_discrete;10}11transformed data {12 vector[N] age30_44; // age as factor13 vector[N] age45_64;14 vector[N] age65up;1516 for (n in 1 : N) {17 age30_44[n] = age_discrete[n] == 2;18 age45_64[n] = age_discrete[n] == 3;19 age65up[n] = age_discrete[n] == 4;20 }21}22parameters {23 vector[9] beta;24 real<lower=0> sigma;25}26model {27 // vectorization28 partyid7 ~ normal(beta[1] + beta[2] * real_ideo + beta[3] * race_adj29 + beta[4] * age30_44 + beta[5] * age45_6430 + beta[6] * age65up + beta[7] * educ1 + beta[8] * gender31 + beta[9] * income, sigma);32}3334//@ DATA { N: 1113, age_discrete: [1113 values], educ1: [1113 values], gender: [1113 values], income: [1113 values], partyid7: [1113 values], race_adj: [1113 values], real_ideo: [1113 values] } // values supplied at runtime35//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","sigma"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0156 ≤ tol 0.0584 · floors 0.0292/0.0186 |
For each of N = 1350 survey respondents, the data provide a party identification score (partyid7), an ideological self-placement score (real_ideo), a race-adjusted value (race_adj), education level (educ1), gender indicator (gender), income level (income), and an age group indicator (age_discrete, an integer in 1, 2, 3, 4 where 1 represents under 30, 2 represents 30-44, 3 represents 45-64, and 4 represents 65+). The regression has nine coefficients: an intercept (beta[1]), a slope for real_ideo (beta[2]), a slope for race_adj (beta[3]), three slopes for age group indicators coded as 30-44 (beta[4]), 45-64 (beta[5]), and 65+ (beta[6]) with under-30 as the reference level, slopes for educ1 (beta[7]), gender (beta[8]), and income (beta[9]). Each regression coefficient has an improper uniform (flat) prior over the reals. The error standard deviation sigma, constrained positive, has an improper uniform prior over the positive reals.
Each respondent's party identification score is normally distributed with a mean equal to the intercept plus the sum of products of slopes and predictors (ideological self-placement, race adjustment, age group indicators, education, gender, and income), and a common standard deviation sigma across all respondents.
The marginal posterior distributions of each of the ten parameters: the intercept (beta[1]), the coefficient for ideological self-placement (beta[2]), the coefficient for race adjustment (beta[3]), the coefficients for the three age group indicators, 30-44 years old (beta[4]), 45-64 years old (beta[5]), and 65 and older (beta[6]), the coefficient for education (beta[7]), the coefficient for gender (beta[8]), the coefficient for income (beta[9]), and the error standard deviation sigma.
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] partyid7;4 vector[N] real_ideo;5 vector[N] race_adj;6 vector[N] educ1;7 vector[N] gender;8 vector[N] income;9 array[N] int age_discrete;10}11transformed data {12 vector[N] age30_44; // age as factor13 vector[N] age45_64;14 vector[N] age65up;1516 for (n in 1 : N) {17 age30_44[n] = age_discrete[n] == 2;18 age45_64[n] = age_discrete[n] == 3;19 age65up[n] = age_discrete[n] == 4;20 }21}22parameters {23 vector[9] beta;24 real<lower=0> sigma;25}26model {27 // vectorization28 partyid7 ~ normal(beta[1] + beta[2] * real_ideo + beta[3] * race_adj29 + beta[4] * age30_44 + beta[5] * age45_6430 + beta[6] * age65up + beta[7] * educ1 + beta[8] * gender31 + beta[9] * income, sigma);32}3334//@ DATA { N: 1350, age_discrete: [1350 values], educ1: [1350 values], gender: [1350 values], income: [1350 values], partyid7: [1350 values], race_adj: [1350 values], real_ideo: [1350 values] } // values supplied at runtime35//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","sigma"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0145 ≤ tol 0.0585 · floors 0.0293/0.0170 |
For each of N respondents, the data provide seven measurements: ideology on the liberal-conservative spectrum (real_ideo), race (race_adj), education level (educ1), gender (binary), income level (income), and age group as an integer in {1, 2, 3, 4}. The regression has nine coefficients: an intercept and eight slopes, corresponding to ideology, race, three age-group indicators (age 30-44, age 45-64, age 65-plus, with age group 1 as the reference category), education, gender, and income. Each coefficient, including the intercept, has a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
For each respondent, the response partyid7 is modeled as normally distributed with a linear predictor and a common standard deviation sigma. The linear predictor is the intercept plus the sum of the eight slope coefficients each multiplied by the corresponding predictor: ideology, race, three binary age-group indicators (which are deterministically constructed from the discrete age variable, with age group 1 as the reference), education, gender, and income. The three age indicators are one if the respondent belongs to that age group and zero otherwise.
The marginal posterior distributions of the nine regression coefficients (the intercept reported as beta[1], and the eight slopes reported as beta[2] through beta[9] for ideology, race, age 30-44, age 45-64, age 65-plus, education, gender, and income respectively) and the error standard deviation sigma.
answer spec
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(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] partyid7;4 vector[N] real_ideo;5 vector[N] race_adj;6 vector[N] educ1;7 vector[N] gender;8 vector[N] income;9 array[N] int age_discrete;10}11transformed data {12 vector[N] age30_44; // age as factor13 vector[N] age45_64;14 vector[N] age65up;1516 for (n in 1 : N) {17 age30_44[n] = age_discrete[n] == 2;18 age45_64[n] = age_discrete[n] == 3;19 age65up[n] = age_discrete[n] == 4;20 }21}22parameters {23 vector[9] beta;24 real<lower=0> sigma;25}26model {27 // vectorization28 partyid7 ~ normal(beta[1] + beta[2] * real_ideo + beta[3] * race_adj29 + beta[4] * age30_44 + beta[5] * age45_6430 + beta[6] * age65up + beta[7] * educ1 + beta[8] * gender31 + beta[9] * income, sigma);32}3334//@ DATA { N: 1043, age_discrete: [1043 values], educ1: [1043 values], gender: [1043 values], income: [1043 values], partyid7: [1043 values], race_adj: [1043 values], real_ideo: [1043 values] } // values supplied at runtime35//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","sigma"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0138 ≤ tol 0.0464 · floors 0.0162/0.0195 |
For each of N = 476 respondents, the data provide the respondent's party identification (on a 7-point scale), ideology score, race adjustment value, education code, gender, income code, and a discrete age group (1, 2, 3, or 4 representing age brackets including one reference age group and three comparison groups). The regression has nine coefficients: an intercept and eight slopes for the predictors (ideology, race adjustment, age groups 2/3/4, education, gender, and income). All nine coefficients have a flat (improper uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a flat (improper uniform) prior over the positive reals.
Each respondent's party identification is normally distributed with a mean equal to the intercept plus the sum of eight slope terms: the ideology coefficient times ideology, the race adjustment coefficient times race adjustment, three age group coefficients for age groups 2, 3, and 4 (with age group 1 as reference), an education coefficient times education, a gender coefficient times gender, and an income coefficient times income. The standard deviation of the response is sigma, common across all respondents.
The marginal posterior distribution of each of the ten parameters: the intercept (reported as beta[1]), the ideology coefficient (reported as beta[2]), the race adjustment coefficient (reported as beta[3]), the three age group coefficients (reported as beta[4], beta[5], beta[6] for age groups 2, 3, and 4 respectively), the education coefficient (reported as beta[7]), the gender coefficient (reported as beta[8]), the income coefficient (reported as beta[9]), and the error standard deviation sigma.
answer spec
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(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] partyid7;4 vector[N] real_ideo;5 vector[N] race_adj;6 vector[N] educ1;7 vector[N] gender;8 vector[N] income;9 array[N] int age_discrete;10}11transformed data {12 vector[N] age30_44; // age as factor13 vector[N] age45_64;14 vector[N] age65up;1516 for (n in 1 : N) {17 age30_44[n] = age_discrete[n] == 2;18 age45_64[n] = age_discrete[n] == 3;19 age65up[n] = age_discrete[n] == 4;20 }21}22parameters {23 vector[9] beta;24 real<lower=0> sigma;25}26model {27 // vectorization28 partyid7 ~ normal(beta[1] + beta[2] * real_ideo + beta[3] * race_adj29 + beta[4] * age30_44 + beta[5] * age45_6430 + beta[6] * age65up + beta[7] * educ1 + beta[8] * gender31 + beta[9] * income, sigma);32}3334//@ DATA { N: 476, age_discrete: [476 values], educ1: [476 values], gender: [476 values], income: [476 values], partyid7: [476 values], race_adj: [476 values], real_ideo: [476 values] } // values supplied at runtime35//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","sigma"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0223 ≤ tol 0.0797 · floors 0.0398/0.0326 |
Survey data from N = 1179 respondents from the National Election Study are provided. For each respondent, the income category income_i (a real-valued predictor, standardized) and the binary vote indicator vote_i (1 = voted Republican, 0 = otherwise) are recorded. The intercept alpha has an implicit improper flat prior (no explicit prior statement in the model). The income coefficient beta[1] has an implicit improper flat prior.
The probability that respondent i voted Republican is inv_logit(alpha + beta[1] * income_i). Each binary vote outcome is Bernoulli with that probability, independently across respondents.
The marginal posterior distribution of each parameter given the data: the intercept alpha and the income coefficient beta[1].
answer spec
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] income;4 array[N] int<lower=0, upper=1> vote;5}6transformed data {7 matrix[N, 1] x = [income']';8}9parameters {10 real alpha;11 vector[1] beta;12}13model {14 vote ~ bernoulli_logit_glm(x, alpha, beta);15}1617//@ DATA { income: [1179 values], N: 1179, vote: [1179 values] } // values supplied at runtime18//@ PARAMS ["alpha","beta[1]"]19//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}20
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0070 ≤ tol 0.0329 · floors 0.0096/0.0144 |
For each of N_t = 20 measurement times, the data provide the observed concentration C_hat in mg/L at that time in days. The pharmacokinetic model is fully specified by four inputs: an initial time t0 (fixed at 0 days), a single administered dose D in mg, and a compartment volume V in liters. The four model parameters are the absorption rate constant k_a (units: 1/day), the Michaelis-Menten half-saturation constant K_m (units: mg/L), the maximum elimination rate V_m (units: 1/day), and the measurement error standard deviation sigma (units: mg/L, all strictly positive). Each of k_a, K_m, V_m, and sigma has a half-Cauchy prior with location 0 and scale 1.
A one-compartment pharmacokinetic model with first-order absorption and saturable Michaelis-Menten elimination. The concentration in the compartment evolves according to an ordinary differential equation. At each time point, the rate of change of concentration depends on two processes: absorption, a first-order term with rate constant k_a that decays exponentially with the administered dose D and compartment volume V; and elimination, a saturable Michaelis-Menten process where the elimination rate equals (V_m / V) times the concentration divided by (K_m plus the concentration), so the rate increases with concentration but approaches the maximum V_m/V asymptotically. The initial concentration at time t0 is zero. For each measurement time, the observed concentration is generated from a lognormal distribution with log-mean equal to the natural logarithm of the ODE-predicted concentration and standard deviation sigma.
The marginal posterior distributions of the four parameters: k_a (the absorption rate constant), K_m (the Michaelis-Menten half-saturation constant), V_m (the maximum elimination rate), and sigma (the measurement error standard deviation).
answer spec
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1functions {2 array[] real one_comp_mm_elim_abs(real t, array[] real y,3 array[] real theta, array[] real x_r,4 array[] int x_i) {5 array[1] real dydt;6 real k_a = theta[1]; // Dosing rate in 1/day7 real K_m = theta[2]; // Michaelis-Menten constant in mg/L8 real V_m = theta[3]; // Maximum elimination rate in 1/day9 real D = x_r[1];10 real V = x_r[2];11 real dose = 0;12 real elim = (V_m / V) * y[1] / (K_m + y[1]);1314 if (t > 0) {15 dose = exp(-k_a * t) * D * k_a / V;16 }1718 dydt[1] = dose - elim;1920 return dydt;21 }22}23data {24 real t0; // Initial time in days;25 // This is currently hardcoded in data transformations26 // Uncomment this line to get original model27 // real C0[1]; // Initial concentration at t0 in mg/L2829 real D; // Total dosage in mg30 real V; // Compartment volume in L3132 int<lower=1> N_t;33 array[N_t] real times; // Measurement times in days3435 // Measured concentrations in effect compartment in mg/L36 array[N_t] real C_hat;37}38transformed data {39 // Comment out the next line to get the original model40 array[1] real C0 = {0.0};41 array[2] real x_r = {D, V};42 array[0] int x_i;43}44parameters {45 real<lower=0> k_a; // Dosing rate in 1/day46 real<lower=0> K_m; // Michaelis-Menten constant in mg/L47 real<lower=0> V_m; // Maximum elimination rate in 1/day48 real<lower=0> sigma;49}50transformed parameters {51 array[N_t, 1] real C;52 {53 array[3] real theta = {k_a, K_m, V_m};54 C = integrate_ode_bdf(one_comp_mm_elim_abs, C0, t0, times, theta, x_r,55 x_i);56 }57}58model {59 // Priors60 k_a ~ cauchy(0, 1);61 K_m ~ cauchy(0, 1);62 V_m ~ cauchy(0, 1);63 sigma ~ cauchy(0, 1);6465 // Likelihood66 for (n in 1 : N_t) {67 C_hat[n] ~ lognormal(log(C[n, 1]), sigma);68 }69}70generated quantities {71 array[N_t] real C_ppc;72 for (n in 1 : N_t) {73 C_ppc[n] = lognormal_rng(log(C[n, 1]), sigma);74 }75}7677//@ DATA { t0: 0, D: 30, V: 2, times: [20 values], N_t: 20, C_hat: [20 values] } // values supplied at runtime78//@ PARAMS ["k_a","K_m","V_m","sigma"]79//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}80
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.4413 ≤ tol 1.8806 · floors 0.9403/0.5151 |
Household radon measurements from 919 homes across 85 Minnesota counties are analyzed. For each measurement i the data provide the log radon level log_radon_i, a floor indicator floor_measure_i (1 if the measurement was taken on the first floor rather than the basement, 0 otherwise), and the county index county_idx_i (1..85). Each county j has its own intercept alpha_j with a Normal(mean 0, sd 10) prior. The floor coefficient beta has a Normal(mean 0, sd 10) prior, and the positive residual standard deviation sigma_y has a half-Normal(0, 1) prior.
The log radon level for measurement i is Normal-distributed with mean alpha[county_idx_i] + beta * floor_measure_i and standard deviation sigma_y.
The marginal posterior distribution of each parameter given the data: the 85 county intercepts alpha_1..alpha_85, the floor coefficient beta, and the residual standard deviation sigma_y.
answer spec
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 int<lower=0> J;4 array[N] int<lower=1, upper=J> county_idx;5 vector[N] floor_measure;6 vector[N] log_radon;7}8parameters {9 vector[J] alpha;10 real beta;11 real<lower=0> sigma_y;12}13model {14 vector[N] mu;15 // Prior16 sigma_y ~ normal(0, 1);17 alpha ~ normal(0, 10);18 beta ~ normal(0, 10);1920 // Likelihood21 for (n in 1 : N) {22 mu[n] = alpha[county_idx[n]] + beta * floor_measure[n];23 target += normal_lpdf(log_radon[n] | mu[n], sigma_y);24 }25}2627//@ DATA { N: 919, J: 85, floor_measure: [919 values], log_radon: [919 values], county_idx: [919 values] } // values supplied at runtime28//@ PARAMS ["alpha[1]","alpha[2]","alpha[3]","alpha[4]","alpha[5]","alpha[6]","alpha[7]","alpha[8]","alpha[9]","alpha[10]","alpha[11]","alpha[12]","alpha[13]","alpha[14]","alpha[15]","alpha[16]","alpha[17]","alpha[18]","alpha[19]","alpha[20]","alpha[21]","alpha[22]","alpha[23]","alpha[24]","alpha[25]","alpha[26]","alpha[27]","alpha[28]","alpha[29]","alpha[30]","alpha[31]","alpha[32]","alpha[33]","alpha[34]","alpha[35]","alpha[36]","alpha[37]","alpha[38]","alpha[39]","alpha[40]","alpha[41]","alpha[42]","alpha[43]","alpha[44]","alpha[45]","alpha[46]","alpha[47]","alpha[48]","alpha[49]","alpha[50]","alpha[51]","alpha[52]","alpha[53]","alpha[54]","alpha[55]","alpha[56]","alpha[57]","alpha[58]","alpha[59]","alpha[60]","alpha[61]","alpha[62]","alpha[63]","alpha[64]","alpha[65]","alpha[66]","alpha[67]","alpha[68]","alpha[69]","alpha[70]","alpha[71]","alpha[72]","alpha[73]","alpha[74]","alpha[75]","alpha[76]","alpha[77]","alpha[78]","alpha[79]","alpha[80]","alpha[81]","alpha[82]","alpha[83]","alpha[84]","alpha[85]","beta","sigma_y"]29//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}30
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| alpha[1] | 0.881 ± 0.370 | — |
| alpha[2] | 0.932 ± 0.097 | — |
| alpha[3] | 1.55 ± 0.426 | — |
| alpha[4] | 1.58 ± 0.284 | — |
| alpha[5] | 1.47 ± 0.362 | — |
| alpha[6] | 1.55 ± 0.423 | — |
| alpha[7] | 2.03 ± 0.190 | — |
| alpha[8] | 1.99 ± 0.367 | — |
| alpha[9] | 1.02 ± 0.236 | — |
| alpha[10] | 1.59 ± 0.290 | — |
| alpha[11] | 1.45 ± 0.333 | — |
| alpha[12] | 1.75 ± 0.358 | — |
| alpha[13] | 1.09 ± 0.289 | — |
| alpha[14] | 2.00 ± 0.190 | — |
| alpha[15] | 1.37 ± 0.369 | — |
| alpha[16] | 0.715 ± 0.495 | — |
| alpha[17] | 1.27 ± 0.368 | — |
| alpha[18] | 1.16 ± 0.219 | — |
| alpha[19] | 1.37 ± 0.090 | — |
| alpha[20] | 1.79 ± 0.411 | — |
| alpha[21] | 1.74 ± 0.248 | — |
| alpha[22] | 0.778 ± 0.317 | — |
| alpha[23] | 1.46 ± 0.520 | — |
| alpha[24] | 2.12 ± 0.250 | — |
| alpha[25] | 1.97 ± 0.189 | — |
| alpha[26] | 1.39 ± 0.070 | — |
| alpha[27] | 1.78 ± 0.301 | — |
| alpha[28] | 1.27 ± 0.336 | — |
| alpha[29] | 1.07 ± 0.413 | — |
| alpha[30] | 0.970 ± 0.219 | — |
| alpha[31] | 2.03 ± 0.327 | — |
| alpha[32] | 1.27 ± 0.370 | — |
| alpha[33] | 2.06 ± 0.369 | — |
| alpha[34] | 1.63 ± 0.410 | — |
| alpha[35] | 0.844 ± 0.276 | — |
| alpha[36] | 2.94 ± 0.511 | — |
| alpha[37] | 0.475 ± 0.245 | — |
| alpha[38] | 1.89 ± 0.373 | — |
| alpha[39] | 1.77 ± 0.321 | — |
| alpha[40] | 2.33 ± 0.349 | — |
| alpha[41] | 1.97 ± 0.252 | — |
| alpha[42] | 1.35 ± 0.729 | — |
| alpha[43] | 1.64 ± 0.238 | — |
| alpha[44] | 1.11 ± 0.263 | — |
| alpha[45] | 1.32 ± 0.194 | — |
| alpha[46] | 1.25 ± 0.308 | — |
| alpha[47] | 0.955 ± 0.496 | — |
| alpha[48] | 1.19 ± 0.242 | — |
| alpha[49] | 1.72 ± 0.192 | — |
| alpha[50] | 2.48 ± 0.742 | — |
| alpha[51] | 2.15 ± 0.367 | — |
| alpha[52] | 1.90 ± 0.415 | — |
| alpha[53] | 1.26 ± 0.439 | — |
| alpha[54] | 1.33 ± 0.144 | — |
| alpha[55] | 1.64 ± 0.266 | — |
| alpha[56] | 1.21 ± 0.417 | — |
| alpha[57] | 0.811 ± 0.302 | — |
| alpha[58] | 1.87 ± 0.377 | — |
| alpha[59] | 1.73 ± 0.376 | — |
| alpha[60] | 1.33 ± 0.506 | — |
| alpha[61] | 1.20 ± 0.128 | — |
| alpha[62] | 2.03 ± 0.323 | — |
| alpha[63] | 1.69 ± 0.408 | — |
| alpha[64] | 1.88 ± 0.215 | — |
| alpha[65] | 1.32 ± 0.497 | — |
| alpha[66] | 1.68 ± 0.197 | — |
| alpha[67] | 1.81 ± 0.209 | — |
| alpha[68] | 1.10 ± 0.254 | — |
| alpha[69] | 1.27 ± 0.356 | — |
| alpha[70] | 0.918 ± 0.072 | — |
| alpha[71] | 1.51 ± 0.146 | — |
| alpha[72] | 1.61 ± 0.225 | — |
| alpha[73] | 1.78 ± 0.531 | — |
| alpha[74] | 1.03 ± 0.366 | — |
| alpha[75] | 1.77 ± 0.420 | — |
| alpha[76] | 2.02 ± 0.348 | — |
| alpha[77] | 1.86 ± 0.278 | — |
| alpha[78] | 1.32 ± 0.318 | — |
| alpha[79] | 0.692 ± 0.355 | — |
| alpha[80] | 1.36 ± 0.112 | — |
| alpha[81] | 2.66 ± 0.412 | — |
| alpha[82] | 2.17 ± 0.721 | — |
| alpha[83] | 1.67 ± 0.206 | — |
| alpha[84] | 1.65 ± 0.198 | — |
| alpha[85] | 1.22 ± 0.498 | — |
| beta | -0.687 ± 0.072 | — |
| sigma_y | 0.727 ± 0.018 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
Household radon measurements from 919 homes across 85 Minnesota counties are analyzed. For each measurement i the data provide the log radon level log_radon_i, a floor indicator floor_measure_i (1 if first floor, 0 if basement), a county-level log uranium predictor log_uppm_i, and the county index county_idx_i (1..85). The county intercepts are partially pooled: their shared mean mu_alpha has a Normal(mean 0, sd 10) prior and their between-county standard deviation sigma_alpha has a half-Normal(0, 1) prior. The two coefficients beta_1 (on log uranium) and beta_2 (on floor) have Normal(mean 0, sd 10) priors, and the positive residual standard deviation sigma_y has a half-Normal(0, 1) prior.
Each county intercept is drawn as alpha_j ~ Normal(mu_alpha, sigma_alpha). The log radon level for measurement i is Normal-distributed with mean alpha[county_idx_i] + log_uppm_i * beta_1 + floor_measure_i * beta_2 and standard deviation sigma_y.
The marginal posterior distribution of each parameter given the data: the 85 county intercepts alpha_1..alpha_85, the uranium coefficient beta_1, the floor coefficient beta_2, the population mean mu_alpha, the between-county standard deviation sigma_alpha, and the residual standard deviation sigma_y.
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"kind": "dist",
"domain": "real"
},
"alpha[43]": {
"kind": "dist",
"domain": "real"
},
"alpha[44]": {
"kind": "dist",
"domain": "real"
},
"alpha[45]": {
"kind": "dist",
"domain": "real"
},
"alpha[46]": {
"kind": "dist",
"domain": "real"
},
"alpha[47]": {
"kind": "dist",
"domain": "real"
},
"alpha[48]": {
"kind": "dist",
"domain": "real"
},
"alpha[49]": {
"kind": "dist",
"domain": "real"
},
"alpha[50]": {
"kind": "dist",
"domain": "real"
},
"alpha[51]": {
"kind": "dist",
"domain": "real"
},
"alpha[52]": {
"kind": "dist",
"domain": "real"
},
"alpha[53]": {
"kind": "dist",
"domain": "real"
},
"alpha[54]": {
"kind": "dist",
"domain": "real"
},
"alpha[55]": {
"kind": "dist",
"domain": "real"
},
"alpha[56]": {
"kind": "dist",
"domain": "real"
},
"alpha[57]": {
"kind": "dist",
"domain": "real"
},
"alpha[58]": {
"kind": "dist",
"domain": "real"
},
"alpha[59]": {
"kind": "dist",
"domain": "real"
},
"alpha[60]": {
"kind": "dist",
"domain": "real"
},
"alpha[61]": {
"kind": "dist",
"domain": "real"
},
"alpha[62]": {
"kind": "dist",
"domain": "real"
},
"alpha[63]": {
"kind": "dist",
"domain": "real"
},
"alpha[64]": {
"kind": "dist",
"domain": "real"
},
"alpha[65]": {
"kind": "dist",
"domain": "real"
},
"alpha[66]": {
"kind": "dist",
"domain": "real"
},
"alpha[67]": {
"kind": "dist",
"domain": "real"
},
"alpha[68]": {
"kind": "dist",
"domain": "real"
},
"alpha[69]": {
"kind": "dist",
"domain": "real"
},
"alpha[70]": {
"kind": "dist",
"domain": "real"
},
"alpha[71]": {
"kind": "dist",
"domain": "real"
},
"alpha[72]": {
"kind": "dist",
"domain": "real"
},
"alpha[73]": {
"kind": "dist",
"domain": "real"
},
"alpha[74]": {
"kind": "dist",
"domain": "real"
},
"alpha[75]": {
"kind": "dist",
"domain": "real"
},
"alpha[76]": {
"kind": "dist",
"domain": "real"
},
"alpha[77]": {
"kind": "dist",
"domain": "real"
},
"alpha[78]": {
"kind": "dist",
"domain": "real"
},
"alpha[79]": {
"kind": "dist",
"domain": "real"
},
"alpha[80]": {
"kind": "dist",
"domain": "real"
},
"alpha[81]": {
"kind": "dist",
"domain": "real"
},
"alpha[82]": {
"kind": "dist",
"domain": "real"
},
"alpha[83]": {
"kind": "dist",
"domain": "real"
},
"alpha[84]": {
"kind": "dist",
"domain": "real"
},
"alpha[85]": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> J;3 int<lower=0> N;4 array[N] int<lower=1, upper=J> county_idx;5 vector[N] log_uppm;6 vector[N] floor_measure;7 vector[N] log_radon;8}9parameters {10 vector[J] alpha_raw;11 vector[2] beta;12 real mu_alpha;13 real<lower=0> sigma_alpha;14 real<lower=0> sigma_y;15}16transformed parameters {17 vector[J] alpha;18 // implies: alpha ~ normal(mu_alpha, sigma_alpha);19 alpha = mu_alpha + sigma_alpha * alpha_raw;20}21model {22 vector[N] mu;23 vector[N] muj;2425 sigma_alpha ~ normal(0, 1);26 sigma_y ~ normal(0, 1);27 mu_alpha ~ normal(0, 10);28 beta ~ normal(0, 10);29 alpha_raw ~ normal(0, 1);3031 for (n in 1 : N) {32 muj[n] = alpha[county_idx[n]] + log_uppm[n] * beta[1];33 mu[n] = muj[n] + floor_measure[n] * beta[2];34 target += normal_lpdf(log_radon[n] | mu[n], sigma_y);35 }36}3738//@ DATA { N: 919, J: 85, floor_measure: [919 values], log_radon: [919 values], log_uppm: [919 values], county_idx: [919 values] } // values supplied at runtime39//@ PARAMS ["beta[1]","beta[2]","mu_alpha","sigma_alpha","sigma_y","alpha[1]","alpha[2]","alpha[3]","alpha[4]","alpha[5]","alpha[6]","alpha[7]","alpha[8]","alpha[9]","alpha[10]","alpha[11]","alpha[12]","alpha[13]","alpha[14]","alpha[15]","alpha[16]","alpha[17]","alpha[18]","alpha[19]","alpha[20]","alpha[21]","alpha[22]","alpha[23]","alpha[24]","alpha[25]","alpha[26]","alpha[27]","alpha[28]","alpha[29]","alpha[30]","alpha[31]","alpha[32]","alpha[33]","alpha[34]","alpha[35]","alpha[36]","alpha[37]","alpha[38]","alpha[39]","alpha[40]","alpha[41]","alpha[42]","alpha[43]","alpha[44]","alpha[45]","alpha[46]","alpha[47]","alpha[48]","alpha[49]","alpha[50]","alpha[51]","alpha[52]","alpha[53]","alpha[54]","alpha[55]","alpha[56]","alpha[57]","alpha[58]","alpha[59]","alpha[60]","alpha[61]","alpha[62]","alpha[63]","alpha[64]","alpha[65]","alpha[66]","alpha[67]","alpha[68]","alpha[69]","alpha[70]","alpha[71]","alpha[72]","alpha[73]","alpha[74]","alpha[75]","alpha[76]","alpha[77]","alpha[78]","alpha[79]","alpha[80]","alpha[81]","alpha[82]","alpha[83]","alpha[84]","alpha[85]"]40//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}41
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| beta[1] | 0.788 ± 0.102 | — |
| beta[2] | -0.643 ± 0.073 | — |
| mu_alpha | 1.42 ± 0.038 | — |
| sigma_alpha | 0.149 ± 0.047 | — |
| sigma_y | 0.730 ± 0.018 | — |
| alpha[1] | 1.41 ± 0.151 | — |
| alpha[2] | 1.42 ± 0.093 | — |
| alpha[3] | 1.44 ± 0.146 | — |
| alpha[4] | 1.52 ± 0.147 | — |
| alpha[5] | 1.43 ± 0.151 | — |
| alpha[6] | 1.39 ± 0.138 | — |
| alpha[7] | 1.53 ± 0.138 | — |
| alpha[8] | 1.46 ± 0.144 | — |
| alpha[9] | 1.35 ± 0.135 | — |
| alpha[10] | 1.41 ± 0.130 | — |
| alpha[11] | 1.49 ± 0.145 | — |
| alpha[12] | 1.42 ± 0.139 | — |
| alpha[13] | 1.43 ± 0.137 | — |
| alpha[14] | 1.51 ± 0.124 | — |
| alpha[15] | 1.41 ± 0.147 | — |
| alpha[16] | 1.38 ± 0.155 | — |
| alpha[17] | 1.34 ± 0.145 | — |
| alpha[18] | 1.45 ± 0.120 | — |
| alpha[19] | 1.34 ± 0.079 | — |
| alpha[20] | 1.44 ± 0.152 | — |
| alpha[21] | 1.46 ± 0.139 | — |
| alpha[22] | 1.23 ± 0.154 | — |
| alpha[23] | 1.39 ± 0.154 | — |
| alpha[24] | 1.54 ± 0.140 | — |
| alpha[25] | 1.53 ± 0.132 | — |
| alpha[26] | 1.39 ± 0.063 | — |
| alpha[27] | 1.40 ± 0.152 | — |
| alpha[28] | 1.42 ± 0.138 | — |
| alpha[29] | 1.44 ± 0.143 | — |
| alpha[30] | 1.39 ± 0.130 | — |
| alpha[31] | 1.49 ± 0.146 | — |
| alpha[32] | 1.39 ± 0.158 | — |
| alpha[33] | 1.49 ± 0.156 | — |
| alpha[34] | 1.45 ± 0.137 | — |
| alpha[35] | 1.41 ± 0.138 | — |
| alpha[36] | 1.52 ± 0.168 | — |
| alpha[37] | 1.27 ± 0.144 | — |
| alpha[38] | 1.54 ± 0.163 | — |
| alpha[39] | 1.45 ± 0.140 | — |
| alpha[40] | 1.49 ± 0.156 | — |
| alpha[41] | 1.47 ± 0.135 | — |
| alpha[42] | 1.42 ± 0.146 | — |
| alpha[43] | 1.45 ± 0.137 | — |
| alpha[44] | 1.30 ± 0.145 | — |
| alpha[45] | 1.32 ± 0.130 | — |
| alpha[46] | 1.37 ± 0.138 | — |
| alpha[47] | 1.39 ± 0.154 | — |
| alpha[48] | 1.35 ± 0.136 | — |
| alpha[49] | 1.45 ± 0.131 | — |
| alpha[50] | 1.46 ± 0.155 | — |
| alpha[51] | 1.50 ± 0.147 | — |
| alpha[52] | 1.42 ± 0.151 | — |
| alpha[53] | 1.37 ± 0.147 | — |
| alpha[54] | 1.27 ± 0.114 | — |
| alpha[55] | 1.49 ± 0.139 | — |
| alpha[56] | 1.40 ± 0.144 | — |
| alpha[57] | 1.31 ± 0.147 | — |
| alpha[58] | 1.44 ± 0.145 | — |
| alpha[59] | 1.42 ± 0.148 | — |
| alpha[60] | 1.39 ± 0.155 | — |
| alpha[61] | 1.41 ± 0.101 | — |
| alpha[62] | 1.46 ± 0.142 | — |
| alpha[63] | 1.42 ± 0.146 | — |
| alpha[64] | 1.50 ± 0.130 | — |
| alpha[65] | 1.38 ± 0.147 | — |
| alpha[66] | 1.53 ± 0.132 | — |
| alpha[67] | 1.51 ± 0.137 | — |
| alpha[68] | 1.44 ± 0.134 | — |
| alpha[69] | 1.37 ± 0.143 | — |
| alpha[70] | 1.23 ± 0.077 | — |
| alpha[71] | 1.39 ± 0.105 | — |
| alpha[72] | 1.39 ± 0.124 | — |
| alpha[73] | 1.42 ± 0.152 | — |
| alpha[74] | 1.30 ± 0.148 | — |
| alpha[75] | 1.45 ± 0.141 | — |
| alpha[76] | 1.44 ± 0.152 | — |
| alpha[77] | 1.47 ± 0.139 | — |
| alpha[78] | 1.47 ± 0.144 | — |
| alpha[79] | 1.28 ± 0.149 | — |
| alpha[80] | 1.40 ± 0.087 | — |
| alpha[81] | 1.54 ± 0.161 | — |
| alpha[82] | 1.44 ± 0.159 | — |
| alpha[83] | 1.35 ± 0.127 | — |
| alpha[84] | 1.50 ± 0.122 | — |
| alpha[85] | 1.37 ± 0.146 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
Household radon measurements from 919 homes across 85 Minnesota counties are analyzed. For each measurement i the data provide the log radon level log_radon_i and the county index county_idx_i (1..85). The county intercepts are partially pooled: their shared mean mu_alpha has a Normal(mean 0, sd 10) prior, their between-county standard deviation sigma_alpha has a half-Normal(0, 1) prior, and the positive residual standard deviation sigma_y has a half-Normal(0, 1) prior.
Each county intercept is drawn as alpha_j ~ Normal(mu_alpha, sigma_alpha). The log radon level for measurement i is Normal-distributed with mean alpha[county_idx_i] and standard deviation sigma_y.
The marginal posterior distribution of each parameter given the data: the 85 county intercepts alpha_1..alpha_85, the population mean mu_alpha, the between-county standard deviation sigma_alpha, and the residual standard deviation sigma_y.
answer spec
{
"kind": "record",
"fields": {
"mu_alpha": {
"kind": "dist",
"domain": "real"
},
"sigma_alpha": {
"kind": "dist",
"domain": "real"
},
"sigma_y": {
"kind": "dist",
"domain": "real"
},
"alpha[1]": {
"kind": "dist",
"domain": "real"
},
"alpha[2]": {
"kind": "dist",
"domain": "real"
},
"alpha[3]": {
"kind": "dist",
"domain": "real"
},
"alpha[4]": {
"kind": "dist",
"domain": "real"
},
"alpha[5]": {
"kind": "dist",
"domain": "real"
},
"alpha[6]": {
"kind": "dist",
"domain": "real"
},
"alpha[7]": {
"kind": "dist",
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},
"alpha[8]": {
"kind": "dist",
"domain": "real"
},
"alpha[9]": {
"kind": "dist",
"domain": "real"
},
"alpha[10]": {
"kind": "dist",
"domain": "real"
},
"alpha[11]": {
"kind": "dist",
"domain": "real"
},
"alpha[12]": {
"kind": "dist",
"domain": "real"
},
"alpha[13]": {
"kind": "dist",
"domain": "real"
},
"alpha[14]": {
"kind": "dist",
"domain": "real"
},
"alpha[15]": {
"kind": "dist",
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},
"alpha[16]": {
"kind": "dist",
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},
"alpha[17]": {
"kind": "dist",
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},
"alpha[18]": {
"kind": "dist",
"domain": "real"
},
"alpha[19]": {
"kind": "dist",
"domain": "real"
},
"alpha[20]": {
"kind": "dist",
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},
"alpha[21]": {
"kind": "dist",
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},
"alpha[22]": {
"kind": "dist",
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},
"alpha[23]": {
"kind": "dist",
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},
"alpha[24]": {
"kind": "dist",
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},
"alpha[25]": {
"kind": "dist",
"domain": "real"
},
"alpha[26]": {
"kind": "dist",
"domain": "real"
},
"alpha[27]": {
"kind": "dist",
"domain": "real"
},
"alpha[28]": {
"kind": "dist",
"domain": "real"
},
"alpha[29]": {
"kind": "dist",
"domain": "real"
},
"alpha[30]": {
"kind": "dist",
"domain": "real"
},
"alpha[31]": {
"kind": "dist",
"domain": "real"
},
"alpha[32]": {
"kind": "dist",
"domain": "real"
},
"alpha[33]": {
"kind": "dist",
"domain": "real"
},
"alpha[34]": {
"kind": "dist",
"domain": "real"
},
"alpha[35]": {
"kind": "dist",
"domain": "real"
},
"alpha[36]": {
"kind": "dist",
"domain": "real"
},
"alpha[37]": {
"kind": "dist",
"domain": "real"
},
"alpha[38]": {
"kind": "dist",
"domain": "real"
},
"alpha[39]": {
"kind": "dist",
"domain": "real"
},
"alpha[40]": {
"kind": "dist",
"domain": "real"
},
"alpha[41]": {
"kind": "dist",
"domain": "real"
},
"alpha[42]": {
"kind": "dist",
"domain": "real"
},
"alpha[43]": {
"kind": "dist",
"domain": "real"
},
"alpha[44]": {
"kind": "dist",
"domain": "real"
},
"alpha[45]": {
"kind": "dist",
"domain": "real"
},
"alpha[46]": {
"kind": "dist",
"domain": "real"
},
"alpha[47]": {
"kind": "dist",
"domain": "real"
},
"alpha[48]": {
"kind": "dist",
"domain": "real"
},
"alpha[49]": {
"kind": "dist",
"domain": "real"
},
"alpha[50]": {
"kind": "dist",
"domain": "real"
},
"alpha[51]": {
"kind": "dist",
"domain": "real"
},
"alpha[52]": {
"kind": "dist",
"domain": "real"
},
"alpha[53]": {
"kind": "dist",
"domain": "real"
},
"alpha[54]": {
"kind": "dist",
"domain": "real"
},
"alpha[55]": {
"kind": "dist",
"domain": "real"
},
"alpha[56]": {
"kind": "dist",
"domain": "real"
},
"alpha[57]": {
"kind": "dist",
"domain": "real"
},
"alpha[58]": {
"kind": "dist",
"domain": "real"
},
"alpha[59]": {
"kind": "dist",
"domain": "real"
},
"alpha[60]": {
"kind": "dist",
"domain": "real"
},
"alpha[61]": {
"kind": "dist",
"domain": "real"
},
"alpha[62]": {
"kind": "dist",
"domain": "real"
},
"alpha[63]": {
"kind": "dist",
"domain": "real"
},
"alpha[64]": {
"kind": "dist",
"domain": "real"
},
"alpha[65]": {
"kind": "dist",
"domain": "real"
},
"alpha[66]": {
"kind": "dist",
"domain": "real"
},
"alpha[67]": {
"kind": "dist",
"domain": "real"
},
"alpha[68]": {
"kind": "dist",
"domain": "real"
},
"alpha[69]": {
"kind": "dist",
"domain": "real"
},
"alpha[70]": {
"kind": "dist",
"domain": "real"
},
"alpha[71]": {
"kind": "dist",
"domain": "real"
},
"alpha[72]": {
"kind": "dist",
"domain": "real"
},
"alpha[73]": {
"kind": "dist",
"domain": "real"
},
"alpha[74]": {
"kind": "dist",
"domain": "real"
},
"alpha[75]": {
"kind": "dist",
"domain": "real"
},
"alpha[76]": {
"kind": "dist",
"domain": "real"
},
"alpha[77]": {
"kind": "dist",
"domain": "real"
},
"alpha[78]": {
"kind": "dist",
"domain": "real"
},
"alpha[79]": {
"kind": "dist",
"domain": "real"
},
"alpha[80]": {
"kind": "dist",
"domain": "real"
},
"alpha[81]": {
"kind": "dist",
"domain": "real"
},
"alpha[82]": {
"kind": "dist",
"domain": "real"
},
"alpha[83]": {
"kind": "dist",
"domain": "real"
},
"alpha[84]": {
"kind": "dist",
"domain": "real"
},
"alpha[85]": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 int<lower=0> J;4 array[N] int<lower=1, upper=J> county_idx;5 vector[N] log_radon;6}7parameters {8 vector[J] alpha_raw;9 real mu_alpha;10 real<lower=0> sigma_alpha;11 real<lower=0> sigma_y;12}13transformed parameters {14 vector[J] alpha;15 // implies: alpha ~ normal(mu_alpha, sigma_alpha);16 alpha = mu_alpha + sigma_alpha * alpha_raw;17}18model {19 vector[N] mu;2021 // priors22 sigma_y ~ normal(0, 1);23 sigma_alpha ~ normal(0, 1);24 mu_alpha ~ normal(0, 10);25 alpha_raw ~ normal(0, 1);2627 // likelihood28 for (n in 1 : N) {29 mu[n] = alpha[county_idx[n]];30 target += normal_lpdf(log_radon[n] | mu[n], sigma_y);31 }32}3334//@ DATA { N: 919, J: 85, log_radon: [919 values], county_idx: [919 values] } // values supplied at runtime35//@ PARAMS ["mu_alpha","sigma_alpha","sigma_y","alpha[1]","alpha[2]","alpha[3]","alpha[4]","alpha[5]","alpha[6]","alpha[7]","alpha[8]","alpha[9]","alpha[10]","alpha[11]","alpha[12]","alpha[13]","alpha[14]","alpha[15]","alpha[16]","alpha[17]","alpha[18]","alpha[19]","alpha[20]","alpha[21]","alpha[22]","alpha[23]","alpha[24]","alpha[25]","alpha[26]","alpha[27]","alpha[28]","alpha[29]","alpha[30]","alpha[31]","alpha[32]","alpha[33]","alpha[34]","alpha[35]","alpha[36]","alpha[37]","alpha[38]","alpha[39]","alpha[40]","alpha[41]","alpha[42]","alpha[43]","alpha[44]","alpha[45]","alpha[46]","alpha[47]","alpha[48]","alpha[49]","alpha[50]","alpha[51]","alpha[52]","alpha[53]","alpha[54]","alpha[55]","alpha[56]","alpha[57]","alpha[58]","alpha[59]","alpha[60]","alpha[61]","alpha[62]","alpha[63]","alpha[64]","alpha[65]","alpha[66]","alpha[67]","alpha[68]","alpha[69]","alpha[70]","alpha[71]","alpha[72]","alpha[73]","alpha[74]","alpha[75]","alpha[76]","alpha[77]","alpha[78]","alpha[79]","alpha[80]","alpha[81]","alpha[82]","alpha[83]","alpha[84]","alpha[85]"]36//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}37
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| mu_alpha | 1.35 ± 0.047 | — |
| sigma_alpha | 0.304 ± 0.047 | — |
| sigma_y | 0.767 ± 0.020 | — |
| alpha[1] | 1.08 ± 0.244 | — |
| alpha[2] | 0.937 ± 0.102 | — |
| alpha[3] | 1.27 ± 0.244 | — |
| alpha[4] | 1.26 ± 0.218 | — |
| alpha[5] | 1.31 ± 0.248 | — |
| alpha[6] | 1.41 ± 0.262 | — |
| alpha[7] | 1.74 ± 0.176 | — |
| alpha[8] | 1.45 ± 0.242 | — |
| alpha[9] | 1.13 ± 0.190 | — |
| alpha[10] | 1.28 ± 0.213 | — |
| alpha[11] | 1.38 ± 0.220 | — |
| alpha[12] | 1.49 ± 0.235 | — |
| alpha[13] | 1.22 ± 0.218 | — |
| alpha[14] | 1.67 ± 0.165 | — |
| alpha[15] | 1.22 ± 0.251 | — |
| alpha[16] | 1.20 ± 0.267 | — |
| alpha[17] | 1.10 ± 0.243 | — |
| alpha[18] | 1.11 ± 0.183 | — |
| alpha[19] | 1.33 ± 0.094 | — |
| alpha[20] | 1.50 ± 0.256 | — |
| alpha[21] | 1.54 ± 0.207 | — |
| alpha[22] | 1.03 ± 0.223 | — |
| alpha[23] | 1.29 ± 0.269 | — |
| alpha[24] | 1.70 ± 0.197 | — |
| alpha[25] | 1.71 ± 0.176 | — |
| alpha[26] | 1.33 ± 0.070 | — |
| alpha[27] | 1.44 ± 0.217 | — |
| alpha[28] | 1.14 ± 0.246 | — |
| alpha[29] | 1.24 ± 0.252 | — |
| alpha[30] | 1.12 ± 0.183 | — |
| alpha[31] | 1.65 ± 0.220 | — |
| alpha[32] | 1.29 ± 0.239 | — |
| alpha[33] | 1.62 ± 0.241 | — |
| alpha[34] | 1.28 ± 0.248 | — |
| alpha[35] | 0.905 ± 0.224 | — |
| alpha[36] | 1.65 ± 0.279 | — |
| alpha[37] | 0.808 ± 0.205 | — |
| alpha[38] | 1.41 ± 0.236 | — |
| alpha[39] | 1.47 ± 0.228 | — |
| alpha[40] | 1.65 ± 0.254 | — |
| alpha[41] | 1.64 ± 0.202 | — |
| alpha[42] | 1.35 ± 0.284 | — |
| alpha[43] | 1.29 ± 0.200 | — |
| alpha[44] | 1.17 ± 0.217 | — |
| alpha[45] | 1.19 ± 0.177 | — |
| alpha[46] | 1.31 ± 0.223 | — |
| alpha[47] | 1.18 ± 0.269 | — |
| alpha[48] | 1.20 ± 0.199 | — |
| alpha[49] | 1.54 ± 0.174 | — |
| alpha[50] | 1.51 ± 0.300 | — |
| alpha[51] | 1.65 ± 0.237 | — |
| alpha[52] | 1.54 ± 0.245 | — |
| alpha[53] | 1.25 ± 0.258 | — |
| alpha[54] | 1.26 ± 0.140 | — |
| alpha[55] | 1.38 ± 0.197 | — |
| alpha[56] | 1.14 ± 0.256 | — |
| alpha[57] | 1.04 ± 0.223 | — |
| alpha[58] | 1.50 ± 0.245 | — |
| alpha[59] | 1.37 ± 0.238 | — |
| alpha[60] | 1.33 ± 0.263 | — |
| alpha[61] | 1.17 ± 0.120 | — |
| alpha[62] | 1.58 ± 0.231 | — |
| alpha[63] | 1.36 ± 0.250 | — |
| alpha[64] | 1.63 ± 0.198 | — |
| alpha[65] | 1.36 ± 0.267 | — |
| alpha[66] | 1.31 ± 0.168 | — |
| alpha[67] | 1.51 ± 0.173 | — |
| alpha[68] | 1.23 ± 0.199 | — |
| alpha[69] | 1.32 ± 0.240 | — |
| alpha[70] | 0.855 ± 0.067 | — |
| alpha[71] | 1.40 ± 0.135 | — |
| alpha[72] | 1.51 ± 0.190 | — |
| alpha[73] | 1.45 ± 0.265 | — |
| alpha[74] | 1.20 ± 0.247 | — |
| alpha[75] | 1.39 ± 0.257 | — |
| alpha[76] | 1.54 ± 0.244 | — |
| alpha[77] | 1.55 ± 0.206 | — |
| alpha[78] | 1.22 ± 0.226 | — |
| alpha[79] | 1.05 ± 0.245 | — |
| alpha[80] | 1.29 ± 0.107 | — |
| alpha[81] | 1.65 ± 0.267 | — |
| alpha[82] | 1.50 ± 0.297 | — |
| alpha[83] | 1.44 ± 0.170 | — |
| alpha[84] | 1.53 ± 0.180 | — |
| alpha[85] | 1.32 ± 0.253 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
Log radon measurements log_radon_i and floor indicators floor_measure_i are available for N = 919 Minnesota homes (i = 1..N), where floor_measure_i = 0 for basement measurements and 1 for first-floor measurements; both arrays are provided as data. The intercept alpha has a Normal(mean 0, sd 10) prior. The floor coefficient beta has a Normal(mean 0, sd 10) prior. The observation noise standard deviation sigma_y is constrained to be positive and has a half-Normal(mean 0, sd 1) prior (a Normal(0, 1) truncated to the positive reals).
The expected log radon for home i is a linear function of its floor indicator: mu_i = alpha + beta * floor_measure_i. The observed log radon log_radon_i is drawn from a Normal distribution with mean mu_i and standard deviation sigma_y, independently across homes.
The marginal posterior distribution of each parameter given the data: the intercept alpha, the floor coefficient beta, and the noise standard deviation sigma_y.
answer spec
{
"kind": "record",
"fields": {
"alpha": {
"kind": "dist",
"domain": "real"
},
"beta": {
"kind": "dist",
"domain": "real"
},
"sigma_y": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] floor_measure;4 vector[N] log_radon;5}6parameters {7 real alpha;8 real beta;9 real<lower=0> sigma_y;10}11model {12 vector[N] mu;1314 // priors15 sigma_y ~ normal(0, 1);16 alpha ~ normal(0, 10);17 beta ~ normal(0, 10);1819 // likelihood20 mu = alpha + beta * floor_measure;21 for (n in 1 : N) {22 target += normal_lpdf(log_radon[n] | mu[n], sigma_y);23 }24}2526//@ DATA { N: 919, floor_measure: [919 values], log_radon: [919 values] } // values supplied at runtime27//@ PARAMS ["alpha","beta","sigma_y"]28//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}29
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0023 ≤ tol 0.0085 · floors 0.0031/0.0032 |
Household radon measurements from 919 homes across 85 Minnesota counties are analyzed. For each measurement i the data provide the log radon level log_radon_i, a floor indicator floor_measure_i (1 if first floor, 0 if basement), and the county index county_idx_i (1..85). The county intercepts are partially pooled: their shared mean mu_alpha has a Normal(mean 0, sd 10) prior and their between-county standard deviation sigma_alpha has a half-Normal(0, 1) prior. The global floor coefficient beta has a Normal(mean 0, sd 10) prior, and the positive residual standard deviation sigma_y has a half-Normal(0, 1) prior.
Each county intercept is drawn as alpha_j ~ Normal(mu_alpha, sigma_alpha). The log radon level for measurement i is Normal-distributed with mean alpha[county_idx_i] + floor_measure_i * beta and standard deviation sigma_y.
The marginal posterior distribution of each parameter given the data: the 85 county intercepts alpha_1..alpha_85, the floor coefficient beta, the population mean mu_alpha, the between-county standard deviation sigma_alpha, and the residual standard deviation sigma_y.
answer spec
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"alpha[84]": {
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"alpha[85]": {
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> J;3 int<lower=0> N;4 array[N] int<lower=1, upper=J> county_idx;5 vector[N] floor_measure;6 vector[N] log_radon;7}8parameters {9 vector[J] alpha_raw;10 real beta;11 real mu_alpha;12 real<lower=0> sigma_alpha;13 real<lower=0> sigma_y;14}15transformed parameters {16 vector[J] alpha;17 // implies: alpha ~ normal(mu_alpha, sigma_alpha);18 alpha = mu_alpha + sigma_alpha * alpha_raw;19}20model {21 vector[N] mu;2223 // Prior24 sigma_y ~ normal(0, 1);25 sigma_alpha ~ normal(0, 1);26 mu_alpha ~ normal(0, 10);27 beta ~ normal(0, 10);28 alpha_raw ~ normal(0, 1);2930 for (n in 1 : N) {31 mu[n] = alpha[county_idx[n]] + floor_measure[n] * beta;32 target += normal_lpdf(log_radon[n] | mu[n], sigma_y);33 }34}3536//@ DATA { N: 919, J: 85, floor_measure: [919 values], log_radon: [919 values], county_idx: [919 values] } // values supplied at runtime37//@ PARAMS ["beta","mu_alpha","sigma_alpha","sigma_y","alpha[1]","alpha[2]","alpha[3]","alpha[4]","alpha[5]","alpha[6]","alpha[7]","alpha[8]","alpha[9]","alpha[10]","alpha[11]","alpha[12]","alpha[13]","alpha[14]","alpha[15]","alpha[16]","alpha[17]","alpha[18]","alpha[19]","alpha[20]","alpha[21]","alpha[22]","alpha[23]","alpha[24]","alpha[25]","alpha[26]","alpha[27]","alpha[28]","alpha[29]","alpha[30]","alpha[31]","alpha[32]","alpha[33]","alpha[34]","alpha[35]","alpha[36]","alpha[37]","alpha[38]","alpha[39]","alpha[40]","alpha[41]","alpha[42]","alpha[43]","alpha[44]","alpha[45]","alpha[46]","alpha[47]","alpha[48]","alpha[49]","alpha[50]","alpha[51]","alpha[52]","alpha[53]","alpha[54]","alpha[55]","alpha[56]","alpha[57]","alpha[58]","alpha[59]","alpha[60]","alpha[61]","alpha[62]","alpha[63]","alpha[64]","alpha[65]","alpha[66]","alpha[67]","alpha[68]","alpha[69]","alpha[70]","alpha[71]","alpha[72]","alpha[73]","alpha[74]","alpha[75]","alpha[76]","alpha[77]","alpha[78]","alpha[79]","alpha[80]","alpha[81]","alpha[82]","alpha[83]","alpha[84]","alpha[85]"]38//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}39
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| beta | -0.663 ± 0.066 | — |
| mu_alpha | 1.49 ± 0.050 | — |
| sigma_alpha | 0.320 ± 0.043 | — |
| sigma_y | 0.728 ± 0.018 | — |
| alpha[1] | 1.22 ± 0.232 | — |
| alpha[2] | 0.988 ± 0.099 | — |
| alpha[3] | 1.52 ± 0.262 | — |
| alpha[4] | 1.54 ± 0.211 | — |
| alpha[5] | 1.46 ± 0.250 | — |
| alpha[6] | 1.49 ± 0.270 | — |
| alpha[7] | 1.86 ± 0.164 | — |
| alpha[8] | 1.70 ± 0.247 | — |
| alpha[9] | 1.20 ± 0.181 | — |
| alpha[10] | 1.53 ± 0.223 | — |
| alpha[11] | 1.45 ± 0.226 | — |
| alpha[12] | 1.59 ± 0.241 | — |
| alpha[13] | 1.27 ± 0.204 | — |
| alpha[14] | 1.87 ± 0.174 | — |
| alpha[15] | 1.43 ± 0.240 | — |
| alpha[16] | 1.29 ± 0.275 | — |
| alpha[17] | 1.40 ± 0.238 | — |
| alpha[18] | 1.26 ± 0.176 | — |
| alpha[19] | 1.38 ± 0.086 | — |
| alpha[20] | 1.60 ± 0.264 | — |
| alpha[21] | 1.66 ± 0.196 | — |
| alpha[22] | 1.12 ± 0.229 | — |
| alpha[23] | 1.49 ± 0.270 | — |
| alpha[24] | 1.88 ± 0.208 | — |
| alpha[25] | 1.84 ± 0.168 | — |
| alpha[26] | 1.40 ± 0.066 | — |
| alpha[27] | 1.64 ± 0.227 | — |
| alpha[28] | 1.36 ± 0.239 | — |
| alpha[29] | 1.35 ± 0.248 | — |
| alpha[30] | 1.15 ± 0.182 | — |
| alpha[31] | 1.74 ± 0.237 | — |
| alpha[32] | 1.38 ± 0.251 | — |
| alpha[33] | 1.75 ± 0.252 | — |
| alpha[34] | 1.53 ± 0.261 | — |
| alpha[35] | 1.13 ± 0.220 | — |
| alpha[36] | 1.89 ± 0.290 | — |
| alpha[37] | 0.860 ± 0.204 | — |
| alpha[38] | 1.63 ± 0.243 | — |
| alpha[39] | 1.62 ± 0.240 | — |
| alpha[40] | 1.84 ± 0.237 | — |
| alpha[41] | 1.78 ± 0.194 | — |
| alpha[42] | 1.47 ± 0.302 | — |
| alpha[43] | 1.58 ± 0.191 | — |
| alpha[44] | 1.28 ± 0.216 | — |
| alpha[45] | 1.36 ± 0.172 | — |
| alpha[46] | 1.39 ± 0.223 | — |
| alpha[47] | 1.36 ± 0.288 | — |
| alpha[48] | 1.29 ± 0.205 | — |
| alpha[49] | 1.67 ± 0.166 | — |
| alpha[50] | 1.68 ± 0.290 | — |
| alpha[51] | 1.79 ± 0.240 | — |
| alpha[52] | 1.66 ± 0.257 | — |
| alpha[53] | 1.40 ± 0.255 | — |
| alpha[54] | 1.37 ± 0.129 | — |
| alpha[55] | 1.59 ± 0.193 | — |
| alpha[56] | 1.37 ± 0.275 | — |
| alpha[57] | 1.12 ± 0.223 | — |
| alpha[58] | 1.67 ± 0.238 | — |
| alpha[59] | 1.61 ± 0.236 | — |
| alpha[60] | 1.44 ± 0.275 | — |
| alpha[61] | 1.25 ± 0.119 | — |
| alpha[62] | 1.75 ± 0.236 | — |
| alpha[63] | 1.56 ± 0.266 | — |
| alpha[64] | 1.73 ± 0.183 | — |
| alpha[65] | 1.47 ± 0.282 | — |
| alpha[66] | 1.61 ± 0.174 | — |
| alpha[67] | 1.72 ± 0.166 | — |
| alpha[68] | 1.28 ± 0.210 | — |
| alpha[69] | 1.41 ± 0.242 | — |
| alpha[70] | 0.942 ± 0.066 | — |
| alpha[71] | 1.50 ± 0.129 | — |
| alpha[72] | 1.57 ± 0.187 | — |
| alpha[73] | 1.59 ± 0.289 | — |
| alpha[74] | 1.28 ± 0.235 | — |
| alpha[75] | 1.58 ± 0.257 | — |
| alpha[76] | 1.73 ± 0.238 | — |
| alpha[77] | 1.69 ± 0.208 | — |
| alpha[78] | 1.41 ± 0.231 | — |
| alpha[79] | 1.13 ± 0.248 | — |
| alpha[80] | 1.38 ± 0.105 | — |
| alpha[81] | 1.94 ± 0.266 | — |
| alpha[82] | 1.61 ± 0.314 | — |
| alpha[83] | 1.61 ± 0.163 | — |
| alpha[84] | 1.60 ± 0.177 | — |
| alpha[85] | 1.44 ± 0.276 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
Household radon measurements from 919 homes across 85 Minnesota counties are analyzed. For each measurement i the data provide the log radon level log_radon_i, a floor indicator floor_measure_i (1 if first floor, 0 if basement), and the county index county_idx_i (1..85). Both the county intercepts and the county floor slopes are partially pooled. The intercept mean mu_alpha and the slope mean mu_beta each have a Normal(mean 0, sd 10) prior. The between-county standard deviations sigma_alpha and sigma_beta, and the residual standard deviation sigma_y, each have a half-Normal(0, 1) prior.
Each county intercept is drawn as alpha_j ~ Normal(mu_alpha, sigma_alpha) and each county floor slope as beta_j ~ Normal(mu_beta, sigma_beta). The log radon level for measurement i is Normal-distributed with mean alpha[county_idx_i] + floor_measure_i * beta[county_idx_i] and standard deviation sigma_y.
The marginal posterior distribution of each parameter given the data: the 85 county intercepts alpha_1..alpha_85, the 85 county floor slopes beta_1..beta_85, the population means mu_alpha and mu_beta, the between-county standard deviations sigma_alpha and sigma_beta, and the residual standard deviation sigma_y.
answer spec
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"domain": "real"
},
"alpha[34]": {
"kind": "dist",
"domain": "real"
},
"alpha[35]": {
"kind": "dist",
"domain": "real"
},
"alpha[36]": {
"kind": "dist",
"domain": "real"
},
"alpha[37]": {
"kind": "dist",
"domain": "real"
},
"alpha[38]": {
"kind": "dist",
"domain": "real"
},
"alpha[39]": {
"kind": "dist",
"domain": "real"
},
"alpha[40]": {
"kind": "dist",
"domain": "real"
},
"alpha[41]": {
"kind": "dist",
"domain": "real"
},
"alpha[42]": {
"kind": "dist",
"domain": "real"
},
"alpha[43]": {
"kind": "dist",
"domain": "real"
},
"alpha[44]": {
"kind": "dist",
"domain": "real"
},
"alpha[45]": {
"kind": "dist",
"domain": "real"
},
"alpha[46]": {
"kind": "dist",
"domain": "real"
},
"alpha[47]": {
"kind": "dist",
"domain": "real"
},
"alpha[48]": {
"kind": "dist",
"domain": "real"
},
"alpha[49]": {
"kind": "dist",
"domain": "real"
},
"alpha[50]": {
"kind": "dist",
"domain": "real"
},
"alpha[51]": {
"kind": "dist",
"domain": "real"
},
"alpha[52]": {
"kind": "dist",
"domain": "real"
},
"alpha[53]": {
"kind": "dist",
"domain": "real"
},
"alpha[54]": {
"kind": "dist",
"domain": "real"
},
"alpha[55]": {
"kind": "dist",
"domain": "real"
},
"alpha[56]": {
"kind": "dist",
"domain": "real"
},
"alpha[57]": {
"kind": "dist",
"domain": "real"
},
"alpha[58]": {
"kind": "dist",
"domain": "real"
},
"alpha[59]": {
"kind": "dist",
"domain": "real"
},
"alpha[60]": {
"kind": "dist",
"domain": "real"
},
"alpha[61]": {
"kind": "dist",
"domain": "real"
},
"alpha[62]": {
"kind": "dist",
"domain": "real"
},
"alpha[63]": {
"kind": "dist",
"domain": "real"
},
"alpha[64]": {
"kind": "dist",
"domain": "real"
},
"alpha[65]": {
"kind": "dist",
"domain": "real"
},
"alpha[66]": {
"kind": "dist",
"domain": "real"
},
"alpha[67]": {
"kind": "dist",
"domain": "real"
},
"alpha[68]": {
"kind": "dist",
"domain": "real"
},
"alpha[69]": {
"kind": "dist",
"domain": "real"
},
"alpha[70]": {
"kind": "dist",
"domain": "real"
},
"alpha[71]": {
"kind": "dist",
"domain": "real"
},
"alpha[72]": {
"kind": "dist",
"domain": "real"
},
"alpha[73]": {
"kind": "dist",
"domain": "real"
},
"alpha[74]": {
"kind": "dist",
"domain": "real"
},
"alpha[75]": {
"kind": "dist",
"domain": "real"
},
"alpha[76]": {
"kind": "dist",
"domain": "real"
},
"alpha[77]": {
"kind": "dist",
"domain": "real"
},
"alpha[78]": {
"kind": "dist",
"domain": "real"
},
"alpha[79]": {
"kind": "dist",
"domain": "real"
},
"alpha[80]": {
"kind": "dist",
"domain": "real"
},
"alpha[81]": {
"kind": "dist",
"domain": "real"
},
"alpha[82]": {
"kind": "dist",
"domain": "real"
},
"alpha[83]": {
"kind": "dist",
"domain": "real"
},
"alpha[84]": {
"kind": "dist",
"domain": "real"
},
"alpha[85]": {
"kind": "dist",
"domain": "real"
},
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"beta[5]": {
"kind": "dist",
"domain": "real"
},
"beta[6]": {
"kind": "dist",
"domain": "real"
},
"beta[7]": {
"kind": "dist",
"domain": "real"
},
"beta[8]": {
"kind": "dist",
"domain": "real"
},
"beta[9]": {
"kind": "dist",
"domain": "real"
},
"beta[10]": {
"kind": "dist",
"domain": "real"
},
"beta[11]": {
"kind": "dist",
"domain": "real"
},
"beta[12]": {
"kind": "dist",
"domain": "real"
},
"beta[13]": {
"kind": "dist",
"domain": "real"
},
"beta[14]": {
"kind": "dist",
"domain": "real"
},
"beta[15]": {
"kind": "dist",
"domain": "real"
},
"beta[16]": {
"kind": "dist",
"domain": "real"
},
"beta[17]": {
"kind": "dist",
"domain": "real"
},
"beta[18]": {
"kind": "dist",
"domain": "real"
},
"beta[19]": {
"kind": "dist",
"domain": "real"
},
"beta[20]": {
"kind": "dist",
"domain": "real"
},
"beta[21]": {
"kind": "dist",
"domain": "real"
},
"beta[22]": {
"kind": "dist",
"domain": "real"
},
"beta[23]": {
"kind": "dist",
"domain": "real"
},
"beta[24]": {
"kind": "dist",
"domain": "real"
},
"beta[25]": {
"kind": "dist",
"domain": "real"
},
"beta[26]": {
"kind": "dist",
"domain": "real"
},
"beta[27]": {
"kind": "dist",
"domain": "real"
},
"beta[28]": {
"kind": "dist",
"domain": "real"
},
"beta[29]": {
"kind": "dist",
"domain": "real"
},
"beta[30]": {
"kind": "dist",
"domain": "real"
},
"beta[31]": {
"kind": "dist",
"domain": "real"
},
"beta[32]": {
"kind": "dist",
"domain": "real"
},
"beta[33]": {
"kind": "dist",
"domain": "real"
},
"beta[34]": {
"kind": "dist",
"domain": "real"
},
"beta[35]": {
"kind": "dist",
"domain": "real"
},
"beta[36]": {
"kind": "dist",
"domain": "real"
},
"beta[37]": {
"kind": "dist",
"domain": "real"
},
"beta[38]": {
"kind": "dist",
"domain": "real"
},
"beta[39]": {
"kind": "dist",
"domain": "real"
},
"beta[40]": {
"kind": "dist",
"domain": "real"
},
"beta[41]": {
"kind": "dist",
"domain": "real"
},
"beta[42]": {
"kind": "dist",
"domain": "real"
},
"beta[43]": {
"kind": "dist",
"domain": "real"
},
"beta[44]": {
"kind": "dist",
"domain": "real"
},
"beta[45]": {
"kind": "dist",
"domain": "real"
},
"beta[46]": {
"kind": "dist",
"domain": "real"
},
"beta[47]": {
"kind": "dist",
"domain": "real"
},
"beta[48]": {
"kind": "dist",
"domain": "real"
},
"beta[49]": {
"kind": "dist",
"domain": "real"
},
"beta[50]": {
"kind": "dist",
"domain": "real"
},
"beta[51]": {
"kind": "dist",
"domain": "real"
},
"beta[52]": {
"kind": "dist",
"domain": "real"
},
"beta[53]": {
"kind": "dist",
"domain": "real"
},
"beta[54]": {
"kind": "dist",
"domain": "real"
},
"beta[55]": {
"kind": "dist",
"domain": "real"
},
"beta[56]": {
"kind": "dist",
"domain": "real"
},
"beta[57]": {
"kind": "dist",
"domain": "real"
},
"beta[58]": {
"kind": "dist",
"domain": "real"
},
"beta[59]": {
"kind": "dist",
"domain": "real"
},
"beta[60]": {
"kind": "dist",
"domain": "real"
},
"beta[61]": {
"kind": "dist",
"domain": "real"
},
"beta[62]": {
"kind": "dist",
"domain": "real"
},
"beta[63]": {
"kind": "dist",
"domain": "real"
},
"beta[64]": {
"kind": "dist",
"domain": "real"
},
"beta[65]": {
"kind": "dist",
"domain": "real"
},
"beta[66]": {
"kind": "dist",
"domain": "real"
},
"beta[67]": {
"kind": "dist",
"domain": "real"
},
"beta[68]": {
"kind": "dist",
"domain": "real"
},
"beta[69]": {
"kind": "dist",
"domain": "real"
},
"beta[70]": {
"kind": "dist",
"domain": "real"
},
"beta[71]": {
"kind": "dist",
"domain": "real"
},
"beta[72]": {
"kind": "dist",
"domain": "real"
},
"beta[73]": {
"kind": "dist",
"domain": "real"
},
"beta[74]": {
"kind": "dist",
"domain": "real"
},
"beta[75]": {
"kind": "dist",
"domain": "real"
},
"beta[76]": {
"kind": "dist",
"domain": "real"
},
"beta[77]": {
"kind": "dist",
"domain": "real"
},
"beta[78]": {
"kind": "dist",
"domain": "real"
},
"beta[79]": {
"kind": "dist",
"domain": "real"
},
"beta[80]": {
"kind": "dist",
"domain": "real"
},
"beta[81]": {
"kind": "dist",
"domain": "real"
},
"beta[82]": {
"kind": "dist",
"domain": "real"
},
"beta[83]": {
"kind": "dist",
"domain": "real"
},
"beta[84]": {
"kind": "dist",
"domain": "real"
},
"beta[85]": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 int<lower=0> J;4 array[N] int<lower=1, upper=J> county_idx;5 vector[N] floor_measure;6 vector[N] log_radon;7}8parameters {9 real<lower=0> sigma_y;10 real<lower=0> sigma_alpha;11 real<lower=0> sigma_beta;12 vector[J] alpha_raw;13 vector[J] beta_raw;14 real mu_alpha;15 real mu_beta;16}17transformed parameters {18 vector[J] alpha;19 vector[J] beta;20 // implies: alpha ~ normal(mu_alpha, sigma_alpha);21 alpha = mu_alpha + sigma_alpha * alpha_raw;22 // implies: beta ~ normal(mu_beta, sigma_beta);23 beta = mu_beta + sigma_beta * beta_raw;24}25model {26 vector[N] mu;27 // Prior28 sigma_y ~ normal(0, 1);29 sigma_beta ~ normal(0, 1);30 sigma_alpha ~ normal(0, 1);31 mu_alpha ~ normal(0, 10);32 mu_beta ~ normal(0, 10);33 alpha_raw ~ normal(0, 1);34 beta_raw ~ normal(0, 1);3536 // Likelihood37 for (n in 1 : N) {38 mu[n] = alpha[county_idx[n]] + floor_measure[n] * beta[county_idx[n]];39 target += normal_lpdf(log_radon[n] | mu[n], sigma_y);40 }41}4243//@ DATA { N: 919, J: 85, floor_measure: [919 values], log_radon: [919 values], county_idx: [919 values] } // values supplied at runtime44//@ PARAMS ["sigma_y","sigma_alpha","sigma_beta","mu_alpha","mu_beta","alpha[1]","alpha[2]","alpha[3]","alpha[4]","alpha[5]","alpha[6]","alpha[7]","alpha[8]","alpha[9]","alpha[10]","alpha[11]","alpha[12]","alpha[13]","alpha[14]","alpha[15]","alpha[16]","alpha[17]","alpha[18]","alpha[19]","alpha[20]","alpha[21]","alpha[22]","alpha[23]","alpha[24]","alpha[25]","alpha[26]","alpha[27]","alpha[28]","alpha[29]","alpha[30]","alpha[31]","alpha[32]","alpha[33]","alpha[34]","alpha[35]","alpha[36]","alpha[37]","alpha[38]","alpha[39]","alpha[40]","alpha[41]","alpha[42]","alpha[43]","alpha[44]","alpha[45]","alpha[46]","alpha[47]","alpha[48]","alpha[49]","alpha[50]","alpha[51]","alpha[52]","alpha[53]","alpha[54]","alpha[55]","alpha[56]","alpha[57]","alpha[58]","alpha[59]","alpha[60]","alpha[61]","alpha[62]","alpha[63]","alpha[64]","alpha[65]","alpha[66]","alpha[67]","alpha[68]","alpha[69]","alpha[70]","alpha[71]","alpha[72]","alpha[73]","alpha[74]","alpha[75]","alpha[76]","alpha[77]","alpha[78]","alpha[79]","alpha[80]","alpha[81]","alpha[82]","alpha[83]","alpha[84]","alpha[85]","beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","beta[10]","beta[11]","beta[12]","beta[13]","beta[14]","beta[15]","beta[16]","beta[17]","beta[18]","beta[19]","beta[20]","beta[21]","beta[22]","beta[23]","beta[24]","beta[25]","beta[26]","beta[27]","beta[28]","beta[29]","beta[30]","beta[31]","beta[32]","beta[33]","beta[34]","beta[35]","beta[36]","beta[37]","beta[38]","beta[39]","beta[40]","beta[41]","beta[42]","beta[43]","beta[44]","beta[45]","beta[46]","beta[47]","beta[48]","beta[49]","beta[50]","beta[51]","beta[52]","beta[53]","beta[54]","beta[55]","beta[56]","beta[57]","beta[58]","beta[59]","beta[60]","beta[61]","beta[62]","beta[63]","beta[64]","beta[65]","beta[66]","beta[67]","beta[68]","beta[69]","beta[70]","beta[71]","beta[72]","beta[73]","beta[74]","beta[75]","beta[76]","beta[77]","beta[78]","beta[79]","beta[80]","beta[81]","beta[82]","beta[83]","beta[84]","beta[85]"]45//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}46
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| sigma_y | 0.721 ± 0.018 | — |
| sigma_alpha | 0.322 ± 0.045 | — |
| sigma_beta | 0.251 ± 0.130 | — |
| mu_alpha | 1.49 ± 0.052 | — |
| mu_beta | -0.652 ± 0.083 | — |
| alpha[1] | 1.22 ± 0.258 | — |
| alpha[2] | 0.994 ± 0.098 | — |
| alpha[3] | 1.49 ± 0.263 | — |
| alpha[4] | 1.55 ± 0.217 | — |
| alpha[5] | 1.47 ± 0.231 | — |
| alpha[6] | 1.49 ± 0.263 | — |
| alpha[7] | 1.85 ± 0.175 | — |
| alpha[8] | 1.67 ± 0.252 | — |
| alpha[9] | 1.18 ± 0.194 | — |
| alpha[10] | 1.52 ± 0.212 | — |
| alpha[11] | 1.45 ± 0.225 | — |
| alpha[12] | 1.61 ± 0.254 | — |
| alpha[13] | 1.26 ± 0.228 | — |
| alpha[14] | 1.88 ± 0.178 | — |
| alpha[15] | 1.43 ± 0.264 | — |
| alpha[16] | 1.27 ± 0.274 | — |
| alpha[17] | 1.43 ± 0.260 | — |
| alpha[18] | 1.22 ± 0.182 | — |
| alpha[19] | 1.39 ± 0.087 | — |
| alpha[20] | 1.61 ± 0.258 | — |
| alpha[21] | 1.64 ± 0.188 | — |
| alpha[22] | 1.10 ± 0.230 | — |
| alpha[23] | 1.44 ± 0.282 | — |
| alpha[24] | 1.88 ± 0.184 | — |
| alpha[25] | 1.81 ± 0.179 | — |
| alpha[26] | 1.40 ± 0.076 | — |
| alpha[27] | 1.65 ± 0.220 | — |
| alpha[28] | 1.37 ± 0.237 | — |
| alpha[29] | 1.34 ± 0.250 | — |
| alpha[30] | 1.14 ± 0.192 | — |
| alpha[31] | 1.76 ± 0.245 | — |
| alpha[32] | 1.38 ± 0.240 | — |
| alpha[33] | 1.76 ± 0.233 | — |
| alpha[34] | 1.51 ± 0.275 | — |
| alpha[35] | 1.10 ± 0.226 | — |
| alpha[36] | 1.88 ± 0.300 | — |
| alpha[37] | 0.854 ± 0.205 | — |
| alpha[38] | 1.64 ± 0.244 | — |
| alpha[39] | 1.63 ± 0.230 | — |
| alpha[40] | 1.87 ± 0.238 | — |
| alpha[41] | 1.78 ± 0.196 | — |
| alpha[42] | 1.48 ± 0.295 | — |
| alpha[43] | 1.68 ± 0.226 | — |
| alpha[44] | 1.27 ± 0.209 | — |
| alpha[45] | 1.38 ± 0.177 | — |
| alpha[46] | 1.34 ± 0.237 | — |
| alpha[47] | 1.36 ± 0.277 | — |
| alpha[48] | 1.28 ± 0.200 | — |
| alpha[49] | 1.68 ± 0.176 | — |
| alpha[50] | 1.66 ± 0.298 | — |
| alpha[51] | 1.78 ± 0.231 | — |
| alpha[52] | 1.65 ± 0.245 | — |
| alpha[53] | 1.42 ± 0.265 | — |
| alpha[54] | 1.38 ± 0.134 | — |
| alpha[55] | 1.55 ± 0.219 | — |
| alpha[56] | 1.41 ± 0.254 | — |
| alpha[57] | 1.12 ± 0.241 | — |
| alpha[58] | 1.68 ± 0.250 | — |
| alpha[59] | 1.58 ± 0.259 | — |
| alpha[60] | 1.43 ± 0.266 | — |
| alpha[61] | 1.22 ± 0.128 | — |
| alpha[62] | 1.71 ± 0.231 | — |
| alpha[63] | 1.55 ± 0.263 | — |
| alpha[64] | 1.73 ± 0.194 | — |
| alpha[65] | 1.44 ± 0.266 | — |
| alpha[66] | 1.61 ± 0.202 | — |
| alpha[67] | 1.68 ± 0.179 | — |
| alpha[68] | 1.27 ± 0.187 | — |
| alpha[69] | 1.40 ± 0.244 | — |
| alpha[70] | 0.936 ± 0.067 | — |
| alpha[71] | 1.51 ± 0.134 | — |
| alpha[72] | 1.57 ± 0.185 | — |
| alpha[73] | 1.61 ± 0.291 | — |
| alpha[74] | 1.27 ± 0.254 | — |
| alpha[75] | 1.57 ± 0.252 | — |
| alpha[76] | 1.73 ± 0.252 | — |
| alpha[77] | 1.71 ± 0.204 | — |
| alpha[78] | 1.43 ± 0.233 | — |
| alpha[79] | 1.13 ± 0.238 | — |
| alpha[80] | 1.39 ± 0.104 | — |
| alpha[81] | 1.88 ± 0.268 | — |
| alpha[82] | 1.60 ± 0.296 | — |
| alpha[83] | 1.65 ± 0.176 | — |
| alpha[84] | 1.62 ± 0.177 | — |
| alpha[85] | 1.43 ± 0.276 | — |
| beta[1] | -0.603 ± 0.278 | — |
| beta[2] | -0.785 ± 0.242 | — |
| beta[3] | -0.635 ± 0.259 | — |
| beta[4] | -0.684 ± 0.238 | — |
| beta[5] | -0.603 ± 0.249 | — |
| beta[6] | -0.670 ± 0.282 | — |
| beta[7] | -0.433 ± 0.310 | — |
| beta[8] | -0.622 ± 0.247 | — |
| beta[9] | -0.506 ± 0.291 | — |
| beta[10] | -0.715 ± 0.242 | — |
| beta[11] | -0.654 ± 0.285 | — |
| beta[12] | -0.654 ± 0.305 | — |
| beta[13] | -0.648 ± 0.310 | — |
| beta[14] | -0.721 ± 0.224 | — |
| beta[15] | -0.646 ± 0.255 | — |
| beta[16] | -0.649 ± 0.311 | — |
| beta[17] | -0.766 ± 0.268 | — |
| beta[18] | -0.520 ± 0.271 | — |
| beta[19] | -0.742 ± 0.228 | — |
| beta[20] | -0.645 ± 0.285 | — |
| beta[21] | -0.604 ± 0.282 | — |
| beta[22] | -0.651 ± 0.258 | — |
| beta[23] | -0.624 ± 0.272 | — |
| beta[24] | -0.633 ± 0.238 | — |
| beta[25] | -0.440 ± 0.285 | — |
| beta[26] | -0.723 ± 0.162 | — |
| beta[27] | -0.606 ± 0.263 | — |
| beta[28] | -0.633 ± 0.235 | — |
| beta[29] | -0.657 ± 0.286 | — |
| beta[30] | -0.630 ± 0.300 | — |
| beta[31] | -0.649 ± 0.299 | — |
| beta[32] | -0.644 ± 0.283 | — |
| beta[33] | -0.653 ± 0.303 | — |
| beta[34] | -0.675 ± 0.260 | — |
| beta[35] | -0.598 ± 0.237 | — |
| beta[36] | -0.510 ± 0.290 | — |
| beta[37] | -0.632 ± 0.259 | — |
| beta[38] | -0.585 ± 0.264 | — |
| beta[39] | -0.584 ± 0.278 | — |
| beta[40] | -0.639 ± 0.253 | — |
| beta[41] | -0.547 ± 0.300 | — |
| beta[42] | -0.674 ± 0.294 | — |
| beta[43] | -0.956 ± 0.286 | — |
| beta[44] | -0.779 ± 0.265 | — |
| beta[45] | -0.750 ± 0.226 | — |
| beta[46] | -0.675 ± 0.298 | — |
| beta[47] | -0.786 ± 0.287 | — |
| beta[48] | -0.598 ± 0.268 | — |
| beta[49] | -0.828 ± 0.272 | — |
| beta[50] | -0.642 ± 0.289 | — |
| beta[51] | -0.654 ± 0.280 | — |
| beta[52] | -0.647 ± 0.287 | — |
| beta[53] | -0.694 ± 0.266 | — |
| beta[54] | -0.825 ± 0.266 | — |
| beta[55] | -0.569 ± 0.243 | — |
| beta[56] | -0.768 ± 0.243 | — |
| beta[57] | -0.677 ± 0.271 | — |
| beta[58] | -0.611 ± 0.259 | — |
| beta[59] | -0.708 ± 0.249 | — |
| beta[60] | -0.639 ± 0.286 | — |
| beta[61] | -0.467 ± 0.259 | — |
| beta[62] | -0.404 ± 0.334 | — |
| beta[63] | -0.553 ± 0.265 | — |
| beta[64] | -0.662 ± 0.266 | — |
| beta[65] | -0.654 ± 0.290 | — |
| beta[66] | -0.625 ± 0.203 | — |
| beta[67] | -0.445 ± 0.259 | — |
| beta[68] | -0.660 ± 0.283 | — |
| beta[69] | -0.649 ± 0.290 | — |
| beta[70] | -0.606 ± 0.153 | — |
| beta[71] | -0.740 ± 0.231 | — |
| beta[72] | -0.658 ± 0.299 | — |
| beta[73] | -0.622 ± 0.301 | — |
| beta[74] | -0.653 ± 0.268 | — |
| beta[75] | -0.562 ± 0.257 | — |
| beta[76] | -0.656 ± 0.252 | — |
| beta[77] | -0.744 ± 0.277 | — |
| beta[78] | -0.707 ± 0.245 | — |
| beta[79] | -0.722 ± 0.251 | — |
| beta[80] | -0.772 ± 0.222 | — |
| beta[81] | -0.476 ± 0.281 | — |
| beta[82] | -0.676 ± 0.307 | — |
| beta[83] | -0.940 ± 0.290 | — |
| beta[84] | -0.667 ± 0.242 | — |
| beta[85] | -0.653 ± 0.289 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
Household radon measurements from 919 homes across 85 Minnesota counties are analyzed. For each measurement i the data provide the log radon level log_radon_i, a floor indicator floor_measure_i (1 if first floor, 0 if basement), and the county index county_idx_i (1..85). The global intercept alpha has a Normal(mean 0, sd 10) prior. The county-varying floor slopes are partially pooled: their shared mean mu_beta has a Normal(mean 0, sd 10) prior, their between-county standard deviation sigma_beta has a half-Normal(0, 1) prior, and the positive residual standard deviation sigma_y has a half-Normal(0, 1) prior.
Each county floor slope is drawn as beta_j ~ Normal(mu_beta, sigma_beta). The log radon level for measurement i is Normal-distributed with mean alpha + floor_measure_i * beta[county_idx_i] and standard deviation sigma_y.
The marginal posterior distribution of each parameter given the data: the global intercept alpha, the 85 county floor slopes beta_1..beta_85, the population slope mean mu_beta, the between-county standard deviation sigma_beta, and the residual standard deviation sigma_y.
answer spec
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}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> J;3 int<lower=0> N;4 array[N] int<lower=1, upper=J> county_idx;5 vector[N] floor_measure;6 vector[N] log_radon;7}8parameters {9 real alpha;10 vector[J] beta_raw;11 real mu_beta;12 real<lower=0> sigma_beta;13 real<lower=0> sigma_y;14}15transformed parameters {16 vector[J] beta;17 // implies: beta ~ normal(mu_beta, sigma_beta);18 beta = mu_beta + sigma_beta * beta_raw;19}20model {21 vector[N] mu;22 // Prior23 alpha ~ normal(0, 10);24 sigma_y ~ normal(0, 1);25 sigma_beta ~ normal(0, 1);26 mu_beta ~ normal(0, 10);27 beta_raw ~ normal(0, 1);2829 for (n in 1 : N) {30 mu[n] = alpha + floor_measure[n] * beta[county_idx[n]];31 target += normal_lpdf(log_radon[n] | mu[n], sigma_y);32 }33}3435//@ DATA { N: 919, J: 85, floor_measure: [919 values], log_radon: [919 values], county_idx: [919 values] } // values supplied at runtime36//@ PARAMS ["alpha","mu_beta","sigma_beta","sigma_y","beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","beta[10]","beta[11]","beta[12]","beta[13]","beta[14]","beta[15]","beta[16]","beta[17]","beta[18]","beta[19]","beta[20]","beta[21]","beta[22]","beta[23]","beta[24]","beta[25]","beta[26]","beta[27]","beta[28]","beta[29]","beta[30]","beta[31]","beta[32]","beta[33]","beta[34]","beta[35]","beta[36]","beta[37]","beta[38]","beta[39]","beta[40]","beta[41]","beta[42]","beta[43]","beta[44]","beta[45]","beta[46]","beta[47]","beta[48]","beta[49]","beta[50]","beta[51]","beta[52]","beta[53]","beta[54]","beta[55]","beta[56]","beta[57]","beta[58]","beta[59]","beta[60]","beta[61]","beta[62]","beta[63]","beta[64]","beta[65]","beta[66]","beta[67]","beta[68]","beta[69]","beta[70]","beta[71]","beta[72]","beta[73]","beta[74]","beta[75]","beta[76]","beta[77]","beta[78]","beta[79]","beta[80]","beta[81]","beta[82]","beta[83]","beta[84]","beta[85]"]37//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}38
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| alpha | 1.36 ± 0.028 | — |
| mu_beta | -0.537 ± 0.086 | — |
| sigma_beta | 0.299 ± 0.115 | — |
| sigma_y | 0.782 ± 0.019 | — |
| beta[1] | -0.562 ± 0.285 | — |
| beta[2] | -0.806 ± 0.265 | — |
| beta[3] | -0.534 ± 0.275 | — |
| beta[4] | -0.536 ± 0.233 | — |
| beta[5] | -0.520 ± 0.299 | — |
| beta[6] | -0.551 ± 0.330 | — |
| beta[7] | -0.216 ± 0.334 | — |
| beta[8] | -0.444 ± 0.256 | — |
| beta[9] | -0.404 ± 0.313 | — |
| beta[10] | -0.600 ± 0.252 | — |
| beta[11] | -0.515 ± 0.333 | — |
| beta[12] | -0.552 ± 0.319 | — |
| beta[13] | -0.541 ± 0.335 | — |
| beta[14] | -0.458 ± 0.245 | — |
| beta[15] | -0.538 ± 0.279 | — |
| beta[16] | -0.556 ± 0.319 | — |
| beta[17] | -0.691 ± 0.260 | — |
| beta[18] | -0.483 ± 0.244 | — |
| beta[19] | -0.649 ± 0.249 | — |
| beta[20] | -0.556 ± 0.338 | — |
| beta[21] | -0.466 ± 0.291 | — |
| beta[22] | -0.595 ± 0.288 | — |
| beta[23] | -0.524 ± 0.293 | — |
| beta[24] | -0.420 ± 0.306 | — |
| beta[25] | -0.207 ± 0.323 | — |
| beta[26] | -0.670 ± 0.180 | — |
| beta[27] | -0.441 ± 0.283 | — |
| beta[28] | -0.577 ± 0.251 | — |
| beta[29] | -0.534 ± 0.334 | — |
| beta[30] | -0.536 ± 0.333 | — |
| beta[31] | -0.503 ± 0.322 | — |
| beta[32] | -0.537 ± 0.321 | — |
| beta[33] | -0.527 ± 0.359 | — |
| beta[34] | -0.551 ± 0.283 | — |
| beta[35] | -0.661 ± 0.235 | — |
| beta[36] | -0.296 ± 0.353 | — |
| beta[37] | -0.601 ± 0.319 | — |
| beta[38] | -0.432 ± 0.276 | — |
| beta[39] | -0.480 ± 0.305 | — |
| beta[40] | -0.467 ± 0.281 | — |
| beta[41] | -0.394 ± 0.331 | — |
| beta[42] | -0.539 ± 0.326 | — |
| beta[43] | -0.822 ± 0.268 | — |
| beta[44] | -0.721 ± 0.298 | — |
| beta[45] | -0.688 ± 0.229 | — |
| beta[46] | -0.540 ± 0.343 | — |
| beta[47] | -0.712 ± 0.297 | — |
| beta[48] | -0.502 ± 0.277 | — |
| beta[49] | -0.675 ± 0.286 | — |
| beta[50] | -0.528 ± 0.329 | — |
| beta[51] | -0.542 ± 0.332 | — |
| beta[52] | -0.533 ± 0.355 | — |
| beta[53] | -0.597 ± 0.293 | — |
| beta[54] | -0.745 ± 0.261 | — |
| beta[55] | -0.457 ± 0.255 | — |
| beta[56] | -0.679 ± 0.268 | — |
| beta[57] | -0.603 ± 0.284 | — |
| beta[58] | -0.453 ± 0.313 | — |
| beta[59] | -0.578 ± 0.281 | — |
| beta[60] | -0.553 ± 0.327 | — |
| beta[61] | -0.404 ± 0.255 | — |
| beta[62] | -0.248 ± 0.346 | — |
| beta[63] | -0.407 ± 0.320 | — |
| beta[64] | -0.537 ± 0.298 | — |
| beta[65] | -0.561 ± 0.337 | — |
| beta[66] | -0.425 ± 0.206 | — |
| beta[67] | -0.250 ± 0.279 | — |
| beta[68] | -0.561 ± 0.341 | — |
| beta[69] | -0.559 ± 0.316 | — |
| beta[70] | -0.835 ± 0.177 | — |
| beta[71] | -0.608 ± 0.247 | — |
| beta[72] | -0.531 ± 0.327 | — |
| beta[73] | -0.542 ± 0.326 | — |
| beta[74] | -0.545 ± 0.335 | — |
| beta[75] | -0.392 ± 0.320 | — |
| beta[76] | -0.503 ± 0.300 | — |
| beta[77] | -0.618 ± 0.294 | — |
| beta[78] | -0.608 ± 0.263 | — |
| beta[79] | -0.637 ± 0.303 | — |
| beta[80] | -0.711 ± 0.242 | — |
| beta[81] | -0.250 ± 0.312 | — |
| beta[82] | -0.555 ± 0.328 | — |
| beta[83] | -0.803 ± 0.279 | — |
| beta[84] | -0.521 ± 0.301 | — |
| beta[85] | -0.538 ± 0.317 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
Thirty young rats had their weights measured at five time points as they aged. The data provide, for each of the 150 measurements, the rat index (1..30), the age x in days, and the measured weight y; the centering age xbar = 22 is also given. Each rat j has its own intercept alpha_j and slope beta_j. The population intercept mean mu_alpha and population slope mean mu_beta each have a Normal(mean 0, sd 100) prior. The three positive standard deviations sigma_y, sigma_alpha, and sigma_beta have improper uniform (flat) priors.
The per-rat intercepts are drawn independently as alpha_j ~ Normal(mu_alpha, sigma_alpha) and the per-rat slopes as beta_j ~ Normal(mu_beta, sigma_beta). Each weight measurement y for rat j at age x is Normal-distributed with mean alpha_j + beta_j * (x - xbar) and standard deviation sigma_y. The intercept at zero age is the derived quantity alpha0 = mu_alpha - xbar * mu_beta.
The marginal posterior distribution of each parameter given the data: the per-rat intercepts alpha_1..alpha_30 and slopes beta_1..beta_30, the population means mu_alpha and mu_beta, the standard deviations sigma_y, sigma_alpha, and sigma_beta, and the derived zero-age intercept alpha0.
answer spec
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"beta[24]": {
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},
"beta[25]": {
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},
"beta[26]": {
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},
"beta[27]": {
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"beta[28]": {
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"beta[29]": {
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"beta[30]": {
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"mu_alpha": {
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"mu_beta": {
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"sigma_y": {
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},
"sigma_alpha": {
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},
"sigma_beta": {
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"alpha0": {
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}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1// Model simplified2data {3 int<lower=0> N; // Number of rats4 int<lower=0> Npts; // Number of data points5 array[Npts] int<lower=0> rat; // Lookup index -> rat6 array[Npts] real x;7 array[Npts] real y;8 real xbar;9}10parameters {11 array[N] real alpha;12 array[N] real beta;1314 real mu_alpha;15 real mu_beta; // beta.c in original bugs model16 real<lower=0> sigma_y; // sigma in original bugs model17 real<lower=0> sigma_alpha;18 real<lower=0> sigma_beta;19}20model {21 mu_alpha ~ normal(0, 100);22 mu_beta ~ normal(0, 100);23 // sigma_y, sigma_alpha, sigma_beta : flat24 alpha ~ normal(mu_alpha, sigma_alpha); // vectorized25 beta ~ normal(mu_beta, sigma_beta); // vectorized26 for (n in 1 : Npts) {27 int irat;28 irat = rat[n];29 y[n] ~ normal(alpha[irat] + beta[irat] * (x[n] - xbar), sigma_y);30 }31}32generated quantities {33 real alpha0;34 alpha0 = mu_alpha - xbar * mu_beta;35}3637//@ DATA { N: 30, Npts: 150, rat: [150 values], x: [150 values], y: [150 values], xbar: 22 } // values supplied at runtime38//@ PARAMS ["alpha[1]","alpha[2]","alpha[3]","alpha[4]","alpha[5]","alpha[6]","alpha[7]","alpha[8]","alpha[9]","alpha[10]","alpha[11]","alpha[12]","alpha[13]","alpha[14]","alpha[15]","alpha[16]","alpha[17]","alpha[18]","alpha[19]","alpha[20]","alpha[21]","alpha[22]","alpha[23]","alpha[24]","alpha[25]","alpha[26]","alpha[27]","alpha[28]","alpha[29]","alpha[30]","beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","beta[6]","beta[7]","beta[8]","beta[9]","beta[10]","beta[11]","beta[12]","beta[13]","beta[14]","beta[15]","beta[16]","beta[17]","beta[18]","beta[19]","beta[20]","beta[21]","beta[22]","beta[23]","beta[24]","beta[25]","beta[26]","beta[27]","beta[28]","beta[29]","beta[30]","mu_alpha","mu_beta","sigma_y","sigma_alpha","sigma_beta","alpha0"]39//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}40
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| alpha[1] | 239.7 ± 2.74 | — |
| alpha[2] | 247.7 ± 2.60 | — |
| alpha[3] | 252.4 ± 2.70 | — |
| alpha[4] | 232.5 ± 2.69 | — |
| alpha[5] | 231.7 ± 2.86 | — |
| alpha[6] | 249.6 ± 2.59 | — |
| alpha[7] | 228.4 ± 2.79 | — |
| alpha[8] | 248.2 ± 2.64 | — |
| alpha[9] | 283.3 ± 2.53 | — |
| alpha[10] | 219.1 ± 2.76 | — |
| alpha[11] | 258.1 ± 2.58 | — |
| alpha[12] | 228.2 ± 2.70 | — |
| alpha[13] | 242.2 ± 2.77 | — |
| alpha[14] | 268.2 ± 2.76 | — |
| alpha[15] | 242.9 ± 2.76 | — |
| alpha[16] | 245.4 ± 2.62 | — |
| alpha[17] | 232.0 ± 2.82 | — |
| alpha[18] | 240.5 ± 2.61 | — |
| alpha[19] | 253.9 ± 2.65 | — |
| alpha[20] | 241.8 ± 2.58 | — |
| alpha[21] | 248.5 ± 2.76 | — |
| alpha[22] | 225.2 ± 2.65 | — |
| alpha[23] | 228.4 ± 2.64 | — |
| alpha[24] | 245.1 ± 2.64 | — |
| alpha[25] | 234.6 ± 2.72 | — |
| alpha[26] | 254.0 ± 2.75 | — |
| alpha[27] | 254.5 ± 2.57 | — |
| alpha[28] | 243.0 ± 2.74 | — |
| alpha[29] | 217.9 ± 2.75 | — |
| alpha[30] | 241.6 ± 2.69 | — |
| beta[1] | 6.06 ± 0.237 | — |
| beta[2] | 7.04 ± 0.252 | — |
| beta[3] | 6.48 ± 0.248 | — |
| beta[4] | 5.34 ± 0.268 | — |
| beta[5] | 6.59 ± 0.247 | — |
| beta[6] | 6.18 ± 0.241 | — |
| beta[7] | 5.98 ± 0.229 | — |
| beta[8] | 6.40 ± 0.242 | — |
| beta[9] | 7.04 ± 0.261 | — |
| beta[10] | 5.86 ± 0.250 | — |
| beta[11] | 6.80 ± 0.252 | — |
| beta[12] | 6.14 ± 0.229 | — |
| beta[13] | 6.16 ± 0.234 | — |
| beta[14] | 6.68 ± 0.242 | — |
| beta[15] | 5.40 ± 0.252 | — |
| beta[16] | 5.93 ± 0.235 | — |
| beta[17] | 6.27 ± 0.249 | — |
| beta[18] | 5.83 ± 0.245 | — |
| beta[19] | 6.42 ± 0.258 | — |
| beta[20] | 6.06 ± 0.248 | — |
| beta[21] | 6.38 ± 0.234 | — |
| beta[22] | 5.85 ± 0.240 | — |
| beta[23] | 5.74 ± 0.252 | — |
| beta[24] | 5.89 ± 0.243 | — |
| beta[25] | 6.91 ± 0.257 | — |
| beta[26] | 6.54 ± 0.253 | — |
| beta[27] | 5.91 ± 0.244 | — |
| beta[28] | 5.83 ± 0.241 | — |
| beta[29] | 5.66 ± 0.247 | — |
| beta[30] | 6.12 ± 0.227 | — |
| mu_alpha | 242.2 ± 2.80 | — |
| mu_beta | 6.19 ± 0.112 | — |
| sigma_y | 6.13 ± 0.464 | — |
| sigma_alpha | 14.85 ± 2.12 | — |
| sigma_beta | 0.530 ± 0.099 | — |
| alpha0 | 106.1 ± 3.59 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
For each of N observations, the data provide D predictors (real-valued) and a response variable (real-valued). The regression has D coefficients, one for each predictor, each with a Normal(0, 10) prior with mean 0 and standard deviation 10. The error standard deviation sigma, constrained positive, has a flat improper uniform prior over the positive reals (equivalent to a normal(0, 10) prior applied to the positive reals).
Each observation's response is Normal-distributed with a mean equal to the linear combination of the D predictors and their coefficients, and a common standard deviation sigma across all observations.
The marginal posterior distributions of each of the D+1 parameters: the D regression coefficients beta[1], beta[2], beta[3], beta[4], beta[5], and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"beta[5]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 int<lower=0> D;4 matrix[N, D] X;5 vector[N] y;6}7parameters {8 vector[D] beta;9 real<lower=0> sigma;10}11model {12 // prior13 target += normal_lpdf(beta | 0, 10);14 target += normal_lpdf(sigma | 0, 10);15 // likelihood16 target += normal_lpdf(y | X * beta, sigma);17}1819//@ DATA { y: [100 values], X: [100×5 matrix], D: 5, N: 100 } // values supplied at runtime20//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","sigma"]21//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}22
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0035 ≤ tol 0.0186 · floors 0.0093/0.0041 |
For each of N = 100 observations, the data provide a response variable y and D = 5 predictor values arranged in a design matrix X (100 rows by 5 columns). The five regression coefficients beta[1], beta[2], beta[3], beta[4], beta[5] each have an improper flat (uniform) prior over the real line. The error standard deviation sigma, constrained positive, has a normal(0, 10) prior.
Each observation's response y[i] is normally distributed with a mean equal to the linear combination of the five predictors weighted by their corresponding coefficients (beta[1] through beta[5]), and a common standard deviation sigma across all observations.
The marginal posterior distributions of the six parameters: the five regression coefficients (beta[1], beta[2], beta[3], beta[4], beta[5]) and the error standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"beta[3]": {
"kind": "dist",
"domain": "real"
},
"beta[4]": {
"kind": "dist",
"domain": "real"
},
"beta[5]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 int<lower=0> D;4 matrix[N, D] X;5 vector[N] y;6}7parameters {8 vector[D] beta;9 real<lower=0> sigma;10}11model {12 // prior13 target += normal_lpdf(beta | 0, 10);14 target += normal_lpdf(sigma | 0, 10);15 // likelihood16 target += normal_lpdf(y | X * beta, sigma);17}1819//@ DATA { y: [100 values], X: [100×5 matrix], D: 5, N: 100 } // values supplied at runtime20//@ PARAMS ["beta[1]","beta[2]","beta[3]","beta[4]","beta[5]","sigma"]21//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}22
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0036 ≤ tol 0.0165 · floors 0.0082/0.0034 |
A seed-germination experiment recorded, for each of 21 plates, the number of seeds that germinated (n) out of the total number planted (N). Two binary plate-level covariates are provided: x1, the seed type, and x2, the root-extract type. The four regression coefficients alpha0, alpha1, alpha2, and alpha12 each have a Normal(mean 0, sd 1000) prior. The precision tau has a Gamma(shape 0.001, rate 0.001) prior, and the random-effect standard deviation is sigma = 1 / sqrt(tau).
Each plate i has a random effect b_i drawn from a Normal(0, sigma) distribution. The number germinated n_i is Binomial with N_i trials and success probability given by the inverse logit of alpha0 + alpha1 * x1_i + alpha2 * x2_i + alpha12 * x1_i * x2_i + b_i.
The marginal posterior distribution of each parameter given the data: the coefficients alpha0, alpha1, alpha2, and alpha12, the per-plate random effects b_1..b_21, the precision tau, and the random-effect standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"alpha0": {
"kind": "dist",
"domain": "real"
},
"alpha1": {
"kind": "dist",
"domain": "real"
},
"alpha12": {
"kind": "dist",
"domain": "real"
},
"alpha2": {
"kind": "dist",
"domain": "real"
},
"tau": {
"kind": "dist",
"domain": "real"
},
"b[1]": {
"kind": "dist",
"domain": "real"
},
"b[2]": {
"kind": "dist",
"domain": "real"
},
"b[3]": {
"kind": "dist",
"domain": "real"
},
"b[4]": {
"kind": "dist",
"domain": "real"
},
"b[5]": {
"kind": "dist",
"domain": "real"
},
"b[6]": {
"kind": "dist",
"domain": "real"
},
"b[7]": {
"kind": "dist",
"domain": "real"
},
"b[8]": {
"kind": "dist",
"domain": "real"
},
"b[9]": {
"kind": "dist",
"domain": "real"
},
"b[10]": {
"kind": "dist",
"domain": "real"
},
"b[11]": {
"kind": "dist",
"domain": "real"
},
"b[12]": {
"kind": "dist",
"domain": "real"
},
"b[13]": {
"kind": "dist",
"domain": "real"
},
"b[14]": {
"kind": "dist",
"domain": "real"
},
"b[15]": {
"kind": "dist",
"domain": "real"
},
"b[16]": {
"kind": "dist",
"domain": "real"
},
"b[17]": {
"kind": "dist",
"domain": "real"
},
"b[18]": {
"kind": "dist",
"domain": "real"
},
"b[19]": {
"kind": "dist",
"domain": "real"
},
"b[20]": {
"kind": "dist",
"domain": "real"
},
"b[21]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> I;3 array[I] int<lower=0> n;4 array[I] int<lower=0> N;5 vector[I] x1;6 vector[I] x2;7}8transformed data {9 vector[I] x1x2;10 x1x2 = x1 .* x2;11}12parameters {13 real alpha0;14 real alpha1;15 real alpha12;16 real alpha2;17 real<lower=0> tau;18 vector[I] b;19}20transformed parameters {21 real<lower=0> sigma;22 sigma = 1.0 / sqrt(tau);23}24model {25 alpha0 ~ normal(0.0, 1.0E3);26 alpha1 ~ normal(0.0, 1.0E3);27 alpha2 ~ normal(0.0, 1.0E3);28 alpha12 ~ normal(0.0, 1.0E3);29 tau ~ gamma(1.0E-3, 1.0E-3);3031 b ~ normal(0.0, sigma);32 n ~ binomial_logit(N,33 alpha0 + alpha1 * x1 + alpha2 * x2 + alpha12 * x1x2 + b);34}3536//@ DATA { I: 21, n: [21 values], N: [21 values], x1: [21 values], x2: [21 values] } // values supplied at runtime37//@ PARAMS ["alpha0","alpha1","alpha12","alpha2","tau","b[1]","b[2]","b[3]","b[4]","b[5]","b[6]","b[7]","b[8]","b[9]","b[10]","b[11]","b[12]","b[13]","b[14]","b[15]","b[16]","b[17]","b[18]","b[19]","b[20]","b[21]","sigma"]38//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}39
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| alpha0 | -0.537 ± 0.198 | — |
| alpha1 | 0.084 ± 0.312 | — |
| alpha12 | -0.829 ± 0.416 | — |
| alpha2 | 1.34 ± 0.275 | — |
| tau | 69.12 ± 187.7 | — |
| b[1] | -0.192 ± 0.258 | — |
| b[2] | 0.011 ± 0.222 | — |
| b[3] | -0.228 ± 0.234 | — |
| b[4] | 0.249 ± 0.252 | — |
| b[5] | 0.089 ± 0.244 | — |
| b[6] | 0.098 ± 0.291 | — |
| b[7] | 0.057 ± 0.216 | — |
| b[8] | 0.177 ± 0.229 | — |
| b[9] | -0.128 ± 0.232 | — |
| b[10] | -0.262 ± 0.254 | — |
| b[11] | 0.091 ± 0.295 | — |
| b[12] | 0.105 ± 0.292 | — |
| b[13] | -0.087 ± 0.235 | — |
| b[14] | -0.134 ± 0.255 | — |
| b[15] | 0.215 ± 0.256 | — |
| b[16] | -0.154 ± 0.320 | — |
| b[17] | -0.223 ± 0.336 | — |
| b[18] | 0.044 ± 0.233 | — |
| b[19] | -0.020 ± 0.228 | — |
| b[20] | 0.213 ± 0.276 | — |
| b[21] | -0.033 ± 0.284 | — |
| sigma | 0.281 ± 0.144 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
A seed-germination experiment recorded, for each of 21 plates, the number of seeds that germinated (n) out of the total number planted (N). Two binary plate-level covariates are provided: x1, the seed type, and x2, the root-extract type. The four regression coefficients alpha0, alpha1, alpha2, and alpha12 each have a Normal(mean 0, sd 1) prior. The random-effect standard deviation sigma, constrained positive, has a half-Cauchy(location 0, scale 1) prior.
Each plate i has a random effect b_i drawn from a Normal(0, sigma) distribution. The number germinated n_i is Binomial with N_i trials and success probability given by the inverse logit of alpha0 + alpha1 * x1_i + alpha2 * x2_i + alpha12 * x1_i * x2_i + b_i.
The marginal posterior distribution of each parameter given the data: the coefficients alpha0, alpha1, alpha2, and alpha12, the per-plate random effects b_1..b_21, and the random-effect standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"alpha0": {
"kind": "dist",
"domain": "real"
},
"alpha1": {
"kind": "dist",
"domain": "real"
},
"alpha12": {
"kind": "dist",
"domain": "real"
},
"alpha2": {
"kind": "dist",
"domain": "real"
},
"b[1]": {
"kind": "dist",
"domain": "real"
},
"b[2]": {
"kind": "dist",
"domain": "real"
},
"b[3]": {
"kind": "dist",
"domain": "real"
},
"b[4]": {
"kind": "dist",
"domain": "real"
},
"b[5]": {
"kind": "dist",
"domain": "real"
},
"b[6]": {
"kind": "dist",
"domain": "real"
},
"b[7]": {
"kind": "dist",
"domain": "real"
},
"b[8]": {
"kind": "dist",
"domain": "real"
},
"b[9]": {
"kind": "dist",
"domain": "real"
},
"b[10]": {
"kind": "dist",
"domain": "real"
},
"b[11]": {
"kind": "dist",
"domain": "real"
},
"b[12]": {
"kind": "dist",
"domain": "real"
},
"b[13]": {
"kind": "dist",
"domain": "real"
},
"b[14]": {
"kind": "dist",
"domain": "real"
},
"b[15]": {
"kind": "dist",
"domain": "real"
},
"b[16]": {
"kind": "dist",
"domain": "real"
},
"b[17]": {
"kind": "dist",
"domain": "real"
},
"b[18]": {
"kind": "dist",
"domain": "real"
},
"b[19]": {
"kind": "dist",
"domain": "real"
},
"b[20]": {
"kind": "dist",
"domain": "real"
},
"b[21]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> I;3 array[I] int<lower=0> n;4 array[I] int<lower=0> N;5 vector[I] x1; // seed type6 vector[I] x2; // root extract7}8transformed data {9 vector[I] x1x2;10 x1x2 = x1 .* x2;11}12parameters {13 real alpha0;14 real alpha1;15 real alpha12;16 real alpha2;17 vector[I] b;18 real<lower=0> sigma;19}20model {21 alpha0 ~ normal(0.0, 1.0); // Narrower priors22 alpha1 ~ normal(0.0, 1.0);23 alpha2 ~ normal(0.0, 1.0);24 alpha12 ~ normal(0.0, 1.0);25 sigma ~ cauchy(0, 1);2627 b ~ normal(0.0, sigma);28 n ~ binomial_logit(N,29 alpha0 + alpha1 * x1 + alpha2 * x2 + alpha12 * x1x2 + b);30}3132//@ DATA { I: 21, n: [21 values], N: [21 values], x1: [21 values], x2: [21 values] } // values supplied at runtime33//@ PARAMS ["alpha0","alpha1","alpha12","alpha2","b[1]","b[2]","b[3]","b[4]","b[5]","b[6]","b[7]","b[8]","b[9]","b[10]","b[11]","b[12]","b[13]","b[14]","b[15]","b[16]","b[17]","b[18]","b[19]","b[20]","b[21]","sigma"]34//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}35
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| alpha0 | -0.448 ± 0.194 | — |
| alpha1 | -0.076 ± 0.303 | — |
| alpha12 | -0.583 ± 0.408 | — |
| alpha2 | 1.19 ± 0.260 | — |
| b[1] | -0.295 ± 0.289 | — |
| b[2] | -0.066 ± 0.243 | — |
| b[3] | -0.318 ± 0.257 | — |
| b[4] | 0.283 ± 0.246 | — |
| b[5] | 0.082 ± 0.235 | — |
| b[6] | 0.133 ± 0.363 | — |
| b[7] | 0.118 ± 0.240 | — |
| b[8] | 0.254 ± 0.256 | — |
| b[9] | -0.111 ± 0.240 | — |
| b[10] | -0.250 ± 0.248 | — |
| b[11] | 0.101 ± 0.320 | — |
| b[12] | 0.192 ± 0.313 | — |
| b[13] | -0.065 ± 0.284 | — |
| b[14] | -0.133 ± 0.280 | — |
| b[15] | 0.308 ± 0.305 | — |
| b[16] | -0.166 ± 0.347 | — |
| b[17] | -0.259 ± 0.333 | — |
| b[18] | 0.043 ± 0.274 | — |
| b[19] | -0.029 ± 0.277 | — |
| b[20] | 0.242 ± 0.280 | — |
| b[21] | -0.059 ± 0.315 | — |
| sigma | 0.344 ± 0.135 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
In a study of the television program Sesame Street, 240 children were each either encouraged or not encouraged to watch the program, and it was recorded whether each child actually watched. For each child i (i = 1..240) two binary indicators are provided as data: encouraged_i, equal to 1 if the child was encouraged and 0 otherwise, and watched_i, equal to 1 if the child watched and 0 otherwise. The regression coefficients beta_1 and beta_2 and the positive residual standard deviation sigma are all given improper uniform (flat) priors.
The watched indicator for child i is modeled as a linear regression on the encouragement indicator: watched_i is Normal-distributed with mean beta_1 + beta_2 * encouraged_i and standard deviation sigma.
The marginal posterior distribution of each parameter given the data: the intercept beta_1, the encouragement coefficient beta_2, and the residual standard deviation sigma.
answer spec
{
"kind": "record",
"fields": {
"beta[1]": {
"kind": "dist",
"domain": "real"
},
"beta[2]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 vector[N] encouraged;4 vector[N] watched;5}6parameters {7 vector[2] beta;8 real<lower=0> sigma;9}10model {11 watched ~ normal(beta[1] + beta[2] * encouraged, sigma);12}1314//@ DATA { watched: [240 values], encouraged: [240 values], N: 240 } // values supplied at runtime15//@ PARAMS ["beta[1]","beta[2]","sigma"]16//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}17
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | — | — |
Surgical mortality records are available for N = 12 hospitals; for each hospital i (i = 1..12) the number of operations n_i and the number of deaths r_i are provided as data arrays. The population mean log-odds mu has a Normal(mean 0, sd 1000) prior. The log-odds variance sigmasq is constrained to be positive and has an InverseGamma(shape 0.001, scale 0.001) prior.
Each hospital has an unknown log-odds of death b_i drawn independently from a Normal distribution with mean mu and standard deviation sigma = sqrt(sigmasq). The hospital's true failure probability is p_i = inv_logit(b_i). The observed number of deaths r_i follows a Binomial distribution with count n_i and probability p_i. A population-level derived quantity is the mean failure probability pop_mean = inv_logit(mu).
The marginal posterior distribution of each parameter and derived quantity given the data: the population log-odds mean mu, the log-odds variance sigmasq, the 12 hospital log-odds b_1, ..., b_12, the log-odds standard deviation sigma, the 12 hospital failure probabilities p_1, ..., p_12, and the population mean failure probability pop_mean.
answer spec
{
"kind": "record",
"fields": {
"mu": {
"kind": "dist",
"domain": "real"
},
"sigmasq": {
"kind": "dist",
"domain": "real"
},
"b[1]": {
"kind": "dist",
"domain": "real"
},
"b[2]": {
"kind": "dist",
"domain": "real"
},
"b[3]": {
"kind": "dist",
"domain": "real"
},
"b[4]": {
"kind": "dist",
"domain": "real"
},
"b[5]": {
"kind": "dist",
"domain": "real"
},
"b[6]": {
"kind": "dist",
"domain": "real"
},
"b[7]": {
"kind": "dist",
"domain": "real"
},
"b[8]": {
"kind": "dist",
"domain": "real"
},
"b[9]": {
"kind": "dist",
"domain": "real"
},
"b[10]": {
"kind": "dist",
"domain": "real"
},
"b[11]": {
"kind": "dist",
"domain": "real"
},
"b[12]": {
"kind": "dist",
"domain": "real"
},
"sigma": {
"kind": "dist",
"domain": "real"
},
"p[1]": {
"kind": "dist",
"domain": "real"
},
"p[2]": {
"kind": "dist",
"domain": "real"
},
"p[3]": {
"kind": "dist",
"domain": "real"
},
"p[4]": {
"kind": "dist",
"domain": "real"
},
"p[5]": {
"kind": "dist",
"domain": "real"
},
"p[6]": {
"kind": "dist",
"domain": "real"
},
"p[7]": {
"kind": "dist",
"domain": "real"
},
"p[8]": {
"kind": "dist",
"domain": "real"
},
"p[9]": {
"kind": "dist",
"domain": "real"
},
"p[10]": {
"kind": "dist",
"domain": "real"
},
"p[11]": {
"kind": "dist",
"domain": "real"
},
"p[12]": {
"kind": "dist",
"domain": "real"
},
"pop_mean": {
"kind": "dist",
"domain": "real"
}
}
}system prompt
(system prompt loads here)
Gold reference posterior draws from posteriordb (10 NUTS chains, R-hat ≈ 1). Not program code — the realized marginals are the answer overlay below.
1data {2 int<lower=0> N;3 array[N] int r;4 array[N] int n;5}6parameters {7 real mu;8 real<lower=0> sigmasq;9 array[N] real b;10}11transformed parameters {12 real<lower=0> sigma;13 array[N] real<lower=0, upper=1> p;14 sigma = sqrt(sigmasq);15 for (i in 1 : N) {16 p[i] = inv_logit(b[i]);17 }18}19model {20 mu ~ normal(0.0, 1000.0);21 sigmasq ~ inv_gamma(0.001, 0.001);22 b ~ normal(mu, sigma);23 r ~ binomial_logit(n, b);24}25generated quantities {26 real pop_mean;27 pop_mean = inv_logit(mu);28}2930//@ DATA { N: 12, n: [12 values], r: [12 values] } // values supplied at runtime31//@ PARAMS ["mu","sigmasq","b[1]","b[2]","b[3]","b[4]","b[5]","b[6]","b[7]","b[8]","b[9]","b[10]","b[11]","b[12]","sigma","p[1]","p[2]","p[3]","p[4]","p[5]","p[6]","p[7]","p[8]","p[9]","p[10]","p[11]","p[12]","pop_mean"]32//@ SAMPLING {"chains":4,"iter_warmup":1000,"iter_sampling":1000}33
| parameter | reference mean±sd | stan mean±sd |
|---|---|---|
| mu | -2.54 ± 0.154 | — |
| sigmasq | 0.186 ± 0.144 | — |
| b[1] | -2.95 ± 0.453 | — |
| b[2] | -2.18 ± 0.231 | — |
| b[3] | -2.61 ± 0.266 | — |
| b[4] | -2.78 ± 0.145 | — |
| b[5] | -2.94 ± 0.277 | — |
| b[6] | -2.60 ± 0.244 | — |
| b[7] | -2.65 ± 0.255 | — |
| b[8] | -1.98 ± 0.213 | — |
| b[9] | -2.61 ± 0.227 | — |
| b[10] | -2.48 ± 0.284 | — |
| b[11] | -2.18 ± 0.192 | — |
| b[12] | -2.63 ± 0.178 | — |
| sigma | 0.405 ± 0.147 | — |
| p[1] | 0.054 ± 0.020 | — |
| p[2] | 0.103 ± 0.022 | — |
| p[3] | 0.070 ± 0.017 | — |
| p[4] | 0.059 ± 0.008 | — |
| p[5] | 0.052 ± 0.013 | — |
| p[6] | 0.071 ± 0.016 | — |
| p[7] | 0.068 ± 0.016 | — |
| p[8] | 0.123 ± 0.023 | — |
| p[9] | 0.070 ± 0.015 | — |
| p[10] | 0.080 ± 0.020 | — |
| p[11] | 0.103 ± 0.018 | — |
| p[12] | 0.068 ± 0.011 | — |
| pop_mean | 0.074 ± 0.010 | — |
| check | status | evidence |
|---|---|---|
| cross-language (stan vs reference) | pass | d=0.0158 ≤ tol 0.0478 · floors 0.0228/0.0239 |