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).

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).

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]).

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.

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.

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).

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.

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.

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.

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).