Modify the factor line in the coin-flipping model so that the soft condition makes heads happen with approximately 95% probability. Use pyro, pyro.distributions as dist, and torch (pre-imported). Enumerate over the discrete latent space manually.
The model: - Sample A from Bernoulli(0.5) (A is 0.0 or 1.0) - Apply a factor of A*3 (i.e., add A*3 to the log-weight) - Return A
Compute the exact posterior distribution over A by enumerating support {0.0, 1.0}, computing unnormalized weights exp(A*3) * prior(A), normalizing, and binding the result to ANSWER as a dict: {"__kind": "distribution", "probs": [...], "support": [false, true]} where support[0]=false corresponds to A=0 and support[1]=true corresponds to A=1.
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1import math23# Prior: flip() = Bernoulli(0.5), support {0, 1}4# Factor: A * 3 added to log weight5# Enumerate support67values = [0.0, 1.0] # false, true8log_prior = math.log(0.5)910weights = []11for a in values:12 log_w = log_prior + a * 3.013 weights.append(math.exp(log_w))1415total = sum(weights)16probs = [w / total for w in weights]1718ANSWER = {"__kind": "distribution", "probs": probs, "support": [False, True]}19
1import torch2import pyro3import pyro.distributions as dist45# Enumerate support {0.0, 1.0}6support = [0.0, 1.0]7prior = dist.Bernoulli(0.5)89log_weights = []10for a in support:11 a_tensor = torch.tensor(a)12 log_prior = prior.log_prob(a_tensor)13 factor = a_tensor * 3.014 log_w = log_prior + factor15 log_weights.append(log_w)1617log_weights_tensor = torch.stack(log_weights)18weights = log_weights_tensor.exp()19weights = weights / weights.sum()2021probs = weights.tolist()2223ANSWER = {24 "__kind": "distribution",25 "probs": probs,26 "support": [False, True]27}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 4.7e-11 | -0.0000 | · |
Three coins are flipped. Use pyro.factor to favor outcomes where the number of heads equals 2. Enumerate all outcomes and compute the posterior marginal distribution over a (the first coin flip). Return the result as a distribution over false/true (represented as support [false, true]).
The model is: - a, b, c each flipped with probability 0.5 - factor: log weight += 1 if (a+b+c)==2, else 0 - return a
pyro, pyro.distributions as dist, and torch are pre-imported. End your program with a top-level assignment ANSWER = {"__kind": "distribution", "probs": [...], "support": [false, true]} where probs are [P(a=False), P(a=True)].
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1import math23import itertools45e = math.e67# Enumerate all 8 outcomes8# a, b, c each 0 or 1 (0=False, 1=True)9total_false = 0.010total_true = 0.01112for a, b, c in itertools.product([0, 1], repeat=3):13 prior = 0.5 ** 3 # 1/814 heads = a + b + c15 # factor(1 * (heads == 2)) means log_weight += 1 if heads==2, else 016 log_factor = 1.0 if heads == 2 else 0.017 weight = prior * math.exp(log_factor)18 if a == 0:19 total_false += weight20 else:21 total_true += weight2223total = total_false + total_true24p_false = total_false / total25p_true = total_true / total2627ANSWER = {"__kind": "distribution", "probs": [p_false, p_true], "support": [False, True]}28
1import itertools23# Enumerate all 8 outcomes for (a, b, c)4outcomes = list(itertools.product([0, 1], repeat=3))56# Compute weights7weights = []8for (a, b, c) in outcomes:9 log_w = 0.0 # prior is uniform, each has prob 0.510 # factor: log weight += 1 if sum == 2, else 011 if a + b + c == 2:12 log_w += 1.013 weights.append(log_w)1415import torch16log_w = torch.tensor(weights)17w = log_w.exp()18w = w / w.sum()1920# Compute marginal over a21p_a_false = sum(w[i] for i, (a, b, c) in enumerate(outcomes) if a == 0)22p_a_true = sum(w[i] for i, (a, b, c) in enumerate(outcomes) if a == 1)2324p_a_false = float(p_a_false)25p_a_true = float(p_a_true)2627ANSWER = {"__kind": "distribution", "probs": [p_a_false, p_a_true], "support": [False, True]}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 3.9e-9 | -0.0000 | · |
Ultimatum game: a proposer allocates $10 between themselves and a responder in $1 increments. The responder accepts any offer >= $1 (strictly greater than 0). The proposer is a soft maximizer: sample offer uniformly from {0,1,...,10}, compute reward = (10 - offer) if responder accepts else 0, apply factor(reward), and return offer.
Using Pyro with pyro, pyro.distributions as dist, and torch pre-imported, compute the exact posterior distribution over the proposer's offer via manual enumeration. Bind the result to a top-level variable ANSWER as a dist.Categorical distribution (with support 0..10). Do not use MCMC.
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1offers = list(range(11))2rewards = []3for offer in offers:4 accepted = offer > 05 reward = (10 - offer) if accepted else 06 rewards.append(float(reward))78import torch9log_weights = torch.tensor(rewards) # factor(reward) means log_weight += reward10weights = log_weights.exp()11weights = weights / weights.sum()12ANSWER = {"__kind": "distribution", "probs": weights.tolist(), "support": offers}
1import torch2import pyro3import pyro.distributions as dist45# Enumerate all possible offers: 0, 1, ..., 106offers = list(range(11))78log_weights = []9for offer in offers:10 # Uniform prior over offers: log(1/11)11 log_prior = torch.tensor(1.0 / 11.0).log()1213 # Responder accepts if offer >= 114 accepts = offer >= 11516 # Reward for proposer17 reward = (10 - offer) if accepts else 01819 # Factor: exp(reward) — soft maximizer20 log_factor = torch.tensor(float(reward))2122 log_weights.append(log_prior + log_factor)2324log_weights = torch.tensor([lw.item() if isinstance(lw, torch.Tensor) else lw for lw in log_weights])25weights = log_weights.exp()26weights = weights / weights.sum()2728ANSWER = dist.Categorical(probs=weights)
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 3.3e-8 | 0.0000 | · |
Implement the modified ultimatum game in Pyro. An alpha = 2 spitefulness parameter governs acceptance: the responder accepts an offer with probability (offer/10)**alpha. Offers are integers 0 through 10 (uniform prior). The reward is (10 - offer) if accepted, else 0. The trace is weighted by exp(reward) (i.e., use pyro.factor with value reward). Enumerate exactly over all offers and responder outcomes to compute the posterior distribution over the proposer's offer. pyro, pyro.distributions as dist, and torch are pre-imported. Bind the result to a top-level variable ANSWER as a dist.Categorical or a dict with __kind, probs, and support keys.
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1import math23alpha = 24offers = list(range(11)) # 0..1056# For each offer o, marginalizing over the responder's flip:7# unnorm_weight(o) = prior(o) * [p * exp(10-o) + (1-p) * exp(0)]8# where p = (o/10)^alpha, prior(o) = 1/11910unnorm_weights = []11for o in offers:12 p = (o / 10) ** alpha13 # factor(reward): reward is (10-o) if accepted, 0 if rejected14 # marginalize over flip15 w = p * math.exp(10 - o) + (1 - p) * math.exp(0)16 unnorm_weights.append(w)1718total = sum(unnorm_weights)19norm_probs = [w / total for w in unnorm_weights]2021ANSWER = {22 "__kind": "distribution",23 "probs": norm_probs,24 "support": offers25}26
1import torch2import pyro3import pyro.distributions as dist45alpha = 26offers = list(range(11)) # 0 through 1078log_weights = []910for offer in offers:11 p_accept = (offer / 10) ** alpha1213 # Two outcomes: accepted (1) or rejected (0)14 # accepted15 reward_accepted = float(10 - offer)16 log_w_accepted = torch.log(torch.tensor(1.0 / 11)) + torch.log(torch.tensor(p_accept)) + torch.tensor(reward_accepted)1718 # rejected19 reward_rejected = 0.020 log_w_rejected = torch.log(torch.tensor(1.0 / 11)) + torch.log(torch.tensor(1.0 - p_accept)) + torch.tensor(reward_rejected)2122 # Combine both outcomes for this offer23 # log sum exp of the two branches24 log_w_offer = torch.logaddexp(log_w_accepted, log_w_rejected)25 log_weights.append(log_w_offer)2627log_weights_tensor = torch.stack(log_weights)28# Normalize29log_weights_tensor = log_weights_tensor - torch.logsumexp(log_weights_tensor, dim=0)30probs = log_weights_tensor.exp()3132ANSWER = dist.Categorical(probs=probs)
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 2.4e-8 | 0.0000 | · |
Implement the Prisoner's Dilemma model in Pyro and bind the result to ANSWER.
Model: Two thieves are interrogated separately. Each independently decides whether to rat with probability 0.5 (Bernoulli flip). Let lenient = 6. The years in jail for the focal thief (IRat) depends on both decisions:
- If other rats and focal rats: years = lenient = 6
- If other rats and focal doesn't: years = 10
- If other doesn't rat and focal rats: years = lenient = 6
- If neither rats: years = 0
Use pyro.factor with log((10 - years) / 10) as the factor weight. Return the posterior distribution over IRat (the focal thief's rat decision, as a boolean or 0/1).
Enumerate all 4 combinations of (otherThiefRats, IRat) ∈ {False, True}² manually, compute unnormalized weights, and construct the posterior.
pyro, pyro.distributions as dist, and torch are pre-imported. Bind the final answer to a top-level variable ANSWER as a dict: {'__kind': 'distribution', 'probs': [...], 'support': [false, true]} (support sorted as [False, True]).
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1import math23lenient = 645# Enumerate all combinations: (otherRats, IRat)6combinations = [(False, False), (False, True), (True, False), (True, True)]78# Prior prob of each combination: 0.259# Compute factor weight for each1011weight_irat_false = 0.012weight_irat_true = 0.01314for (otherRats, IRat) in combinations:15 if otherRats:16 years = lenient if IRat else 1017 else:18 years = lenient if IRat else 019 percent_freedom = (10 - years) / 1020 # factor weight is percent_freedom (in log space: log(percent_freedom))21 # prior prob is 0.2522 w = 0.25 * percent_freedom23 if IRat:24 weight_irat_true += w25 else:26 weight_irat_false += w2728total = weight_irat_false + weight_irat_true29prob_false = weight_irat_false / total30prob_true = weight_irat_true / total3132ANSWER = {33 "__kind": "distribution",34 "probs": [prob_false, prob_true],35 "support": [False, True]36}37
1import itertools2import math34# Enumerate all 4 combinations of (otherThiefRats, IRat)5# otherThiefRats in {False, True}, IRat in {False, True}6# Each has prior probability 0.5 * 0.5 = 0.2578lenient = 6910# Compute unnormalized weights for each value of IRat11# We marginalize over otherThiefRats1213# For IRat = False (0):14# otherThiefRats = False: years = 0, factor = log((10-0)/10) = log(1) = 015# otherThiefRats = True: years = 10, factor = log((10-10)/10) = log(0) = -inf16# For IRat = True (1):17# otherThiefRats = False: years = lenient = 6, factor = log((10-6)/10) = log(0.4)18# otherThiefRats = True: years = lenient = 6, factor = log((10-6)/10) = log(0.4)1920log_weights = {False: [], True: []}2122for otherRats in [False, True]:23 for iRat in [False, True]:24 if otherRats and iRat:25 years = lenient # 626 elif otherRats and not iRat:27 years = 1028 elif not otherRats and iRat:29 years = lenient # 630 else: # neither rats31 years = 03233 factor_val = (10 - years) / 1034 if factor_val <= 0:35 log_factor = float('-inf')36 else:37 log_factor = math.log(factor_val)3839 # Prior: 0.5 * 0.5 = 0.25 for each combination40 log_prior = math.log(0.25)41 log_w = log_prior + log_factor42 log_weights[iRat].append(log_w)4344# Sum weights for each value of IRat (marginalize over otherThiefRats)45import torch4647def logsumexp(vals):48 max_v = max(vals)49 if max_v == float('-inf'):50 return float('-inf')51 return max_v + math.log(sum(math.exp(v - max_v) for v in vals))5253log_w_false = logsumexp(log_weights[False])54log_w_true = logsumexp(log_weights[True])5556# Normalize57max_lw = max(log_w_false, log_w_true)58w_false = math.exp(log_w_false - max_lw)59w_true = math.exp(log_w_true - max_lw)60total = w_false + w_true61p_false = w_false / total62p_true = w_true / total6364ANSWER = {65 '__kind': 'distribution',66 'probs': [p_false, p_true],67 'support': [False, True]68}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 8.3e-17 | -0.0000 | · |
Implement the Frank & Goodman RSA model in Pyro. The world has three objects: {shape:'square',color:'blue'}, {shape:'circle',color:'blue'}, {shape:'square',color:'green'}, drawn uniformly. Possible utterances: ['blue','green','square','circle']. Truth function: color/shape utterances must match the corresponding attribute.
Build: - literalListener(utterance): posterior over objects given utterance is true - speaker(obj, alpha): distribution over utterances, scored by alpha * literalListener(utterance).log_prob(obj) - pragmaticListener(utterance, alpha): posterior over objects given speaker chose utterance
Compute pragmaticListener('blue', alpha) for four alpha values analytically via enumeration. Return a dict named ANSWER with keys 'alpha_001', 'alpha_1', 'alpha_4', 'alpha_10', each value being a dict with '__kind':'distribution', 'probs':[...], 'support':[...] where support items are dicts {shape:..., color:...} sorted canonically. Note: pyro, pyro.distributions as dist, and torch are pre-imported.
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1import math23objects = [4 {"shape": "square", "color": "blue"},5 {"shape": "circle", "color": "blue"},6 {"shape": "square", "color": "green"},7]89utterances = ["blue", "green", "square", "circle"]1011def meaning(utterance, obj):12 if utterance in ("blue", "green"):13 return utterance == obj["color"]14 elif utterance in ("circle", "square"):15 return utterance == obj["shape"]16 else:17 return True1819# Literal listener: P(obj | utterance) proportional to meaning * uniform prior20def literal_listener(utterance):21 scores = [1.0 if meaning(utterance, obj) else 0.0 for obj in objects]22 total = sum(scores)23 return [s / total for s in scores] # list indexed by objects2425# Speaker: P(utt | obj, alpha) proportional to exp(alpha * log P_L0(obj | utt))26def speaker(obj_idx, alpha):27 obj = objects[obj_idx]28 log_weights = []29 for utt in utterances:30 ll = literal_listener(utt)31 p = ll[obj_idx]32 if p > 0:33 log_weights.append(alpha * math.log(p))34 else:35 log_weights.append(float('-inf'))36 # normalize37 max_lw = max(lw for lw in log_weights if lw != float('-inf'))38 weights = [math.exp(lw - max_lw) if lw != float('-inf') else 0.0 for lw in log_weights]39 total = sum(weights)40 return [w / total for w in weights] # prob over utterances4142def pragmatic_listener(utterance, alpha):43 utt_idx = utterances.index(utterance)44 # prior is uniform over objects45 weights = []46 for obj_idx in range(len(objects)):47 sp = speaker(obj_idx, alpha)48 weights.append((1.0/3.0) * sp[utt_idx])49 total = sum(weights)50 probs = [w / total for w in weights]51 return probs # indexed by objects5253def make_dist(utterance, alpha):54 probs = pragmatic_listener(utterance, alpha)55 # filter to nonzero support56 support = []57 filtered_probs = []58 for i, obj in enumerate(objects):59 if probs[i] > 0:60 support.append(obj)61 filtered_probs.append(probs[i])62 # sort support canonically (by JSON repr)63 import json64 combined = sorted(zip(support, filtered_probs), key=lambda x: json.dumps(x[0], sort_keys=True))65 support_sorted = [c[0] for c in combined]66 probs_sorted = [c[1] for c in combined]67 return {"__kind": "distribution", "probs": probs_sorted, "support": support_sorted}6869ANSWER = {70 "alpha_001": make_dist("blue", 0.01),71 "alpha_1": make_dist("blue", 1),72 "alpha_4": make_dist("blue", 4),73 "alpha_10": make_dist("blue", 10),74}75
1import torch2import pyro3import pyro.distributions as dist45# World objects6objects = [7 {'shape': 'square', 'color': 'blue'},8 {'shape': 'circle', 'color': 'blue'},9 {'shape': 'square', 'color': 'green'},10]1112# Utterances13utterances = ['blue', 'green', 'square', 'circle']1415# Truth function16def meaning(utterance, obj):17 if utterance == 'blue':18 return obj['color'] == 'blue'19 elif utterance == 'green':20 return obj['color'] == 'green'21 elif utterance == 'square':22 return obj['shape'] == 'square'23 elif utterance == 'circle':24 return obj['shape'] == 'circle'25 return False2627# Literal listener: posterior over objects given utterance28def literal_listener(utterance):29 # Uniform prior over objects, conditioned on meaning30 probs = []31 for obj in objects:32 probs.append(1.0 if meaning(utterance, obj) else 0.0)33 probs = torch.tensor(probs)34 probs = probs / probs.sum()35 return probs # distribution over objects indices 0,1,23637# Speaker: distribution over utterances given object38def speaker_probs(obj_idx, alpha):39 # For each utterance, compute alpha * log P_L0(obj | utterance)40 scores = []41 for utt in utterances:42 ll_probs = literal_listener(utt)43 log_prob = torch.log(ll_probs[obj_idx] + 1e-30)44 scores.append(alpha * log_prob)45 scores = torch.tensor(scores)46 # Softmax47 scores = scores - scores.max()48 exp_scores = scores.exp()49 return exp_scores / exp_scores.sum()5051# Pragmatic listener: posterior over objects given utterance52def pragmatic_listener(utterance, alpha):53 utt_idx = utterances.index(utterance)54 # P_L1(obj | utt) proportional to P_S1(utt | obj) * P(obj)55 scores = []56 for obj_idx in range(len(objects)):57 sp = speaker_probs(obj_idx, alpha)58 # uniform prior over objects59 scores.append(sp[utt_idx] * (1.0 / len(objects)))60 scores = torch.stack(scores)61 scores = scores / scores.sum()62 return scores6364# Compute for four alpha values65alpha_values = {66 'alpha_001': 0.01,67 'alpha_1': 1.0,68 'alpha_4': 4.0,69 'alpha_10': 10.0,70}7172support = [{'shape': obj['shape'], 'color': obj['color']} for obj in objects]7374ANSWER = {}75for key, alpha in alpha_values.items():76 probs = pragmatic_listener('blue', alpha)77 ANSWER[key] = {78 '__kind': 'distribution',79 'probs': probs.tolist(),80 'support': support,81 }
| run | flags | bucket | TV | KL | actions |
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◐TV<.5 | 0.201 | 4.1331 | · |
Implement the RSA (Rational Speech Acts) model in Pyro. The world has three objects: {shape: 'square', color: 'blue'}, {shape: 'circle', color: 'blue'}, {shape: 'square', color: 'green'}, drawn uniformly. Possible utterances: ['blue', 'green', 'square', 'circle']. Truth function: a color utterance ('blue'/'green') is true iff it matches obj.color; a shape utterance ('circle'/'square') is true iff it matches obj.shape.
Build level-1 RSA with alpha=1: - literalListener(utt): uniform prior over objects, conditioned on meaning(utt, obj) being true. - speaker(obj): uniform prior over utterances, weighted by exp(alpha * literalListener(utt).log_prob(obj)). - pragmaticListener(utt) = L1: uniform prior over objects, weighted by speaker(obj).log_prob(utt).
Build level-2 RSA with alpha=1: - speaker2(obj): uniform prior over utterances, weighted by exp(alpha * pragmaticListener(utt).log_prob(obj)). - listener3(utt) = L2: uniform prior over objects, weighted by speaker2(obj).log_prob(utt).
Compute L1 = pragmaticListener('blue') and L2 = listener3('blue'). Both should be distributions over objects.
Bind the result as: ANSWER = {'L1': {'__kind': 'distribution', 'probs': [...], 'support': [...]}, 'L2': {...}}
where support items are dicts with 'shape' and 'color' keys, sorted canonically (alphabetical by JSON serialization). Note: pyro, pyro.distributions as dist, and torch are pre-imported.
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1import math23objects = [4 {"shape": "square", "color": "blue"},5 {"shape": "circle", "color": "blue"},6 {"shape": "square", "color": "green"}7]89utterances = ["blue", "green", "square", "circle"]10alpha = 1.01112def meaning(utt, obj):13 if utt in ("blue", "green"):14 return utt == obj["color"]15 elif utt in ("circle", "square"):16 return utt == obj["shape"]17 return True1819# Literal listener: p(obj|utt) proportional to meaning(utt,obj) * uniform prior20def literal_listener(utt):21 scores = []22 for obj in objects:23 scores.append(1.0 if meaning(utt, obj) else 0.0)24 total = sum(scores)25 return [s / total for s in scores]2627# Speaker: p(utt|obj) proportional to exp(alpha * log L0(obj|utt))28def speaker(obj):29 obj_idx = objects.index(obj)30 scores = []31 for utt in utterances:32 ll = literal_listener(utt)33 p_obj = ll[obj_idx]34 if p_obj > 0:35 scores.append(math.exp(alpha * math.log(p_obj)))36 else:37 scores.append(0.0)38 total = sum(scores)39 return [s / total for s in scores]4041# Pragmatic listener L1: p(obj|utt) proportional to speaker(obj, utt) * prior42def pragmatic_listener(utt):43 utt_idx = utterances.index(utt)44 scores = []45 for obj in objects:46 sp = speaker(obj)47 scores.append(sp[utt_idx] * (1.0 / len(objects)))48 total = sum(scores)49 return [s / total for s in scores]5051# Speaker2: p(utt|obj) proportional to exp(alpha * log L1(obj|utt))52def speaker2(obj):53 obj_idx = objects.index(obj)54 scores = []55 for utt in utterances:56 pl = pragmatic_listener(utt)57 p_obj = pl[obj_idx]58 if p_obj > 0:59 scores.append(math.exp(alpha * math.log(p_obj)))60 else:61 scores.append(0.0)62 total = sum(scores)63 return [s / total for s in scores]6465# Listener3 L2: p(obj|utt) proportional to speaker2(obj, utt) * prior66def listener3(utt):67 utt_idx = utterances.index(utt)68 scores = []69 for obj in objects:70 sp2 = speaker2(obj)71 scores.append(sp2[utt_idx] * (1.0 / len(objects)))72 total = sum(scores)73 return [s / total for s in scores]7475# Compute L1 and L2 for utterance 'blue'76l1_probs = pragmatic_listener("blue")77l2_probs = listener3("blue")7879# Build support sorted canonically (JSON serialization order: color then shape alphabetically)80import json8182def obj_key(obj):83 return json.dumps(obj, sort_keys=True)8485# Filter to non-zero probability objects and sort86l1_pairs = [(obj, p) for obj, p in zip(objects, l1_probs) if p > 0]87l1_pairs_sorted = sorted(l1_pairs, key=lambda x: obj_key(x[0]))8889l2_pairs = [(obj, p) for obj, p in zip(objects, l2_probs) if p > 0]90l2_pairs_sorted = sorted(l2_pairs, key=lambda x: obj_key(x[0]))9192ANSWER = {93 "L1": {94 "__kind": "distribution",95 "probs": [p for _, p in l1_pairs_sorted],96 "support": [o for o, _ in l1_pairs_sorted]97 },98 "L2": {99 "__kind": "distribution",100 "probs": [p for _, p in l2_pairs_sorted],101 "support": [o for o, _ in l2_pairs_sorted]102 }103}104
1import torch2import itertools34# Define objects5objects = [6 {'shape': 'square', 'color': 'blue'},7 {'shape': 'circle', 'color': 'blue'},8 {'shape': 'square', 'color': 'green'},9]1011# Utterances12utterances = ['blue', 'green', 'square', 'circle']1314alpha = 1.01516# Truth function17def meaning(utt, obj):18 if utt in ['blue', 'green']:19 return obj['color'] == utt20 elif utt in ['square', 'circle']:21 return obj['shape'] == utt22 return False2324# Literal listener: P(obj | utt) proportional to meaning(utt, obj) * uniform prior25def literal_listener(utt):26 weights = torch.tensor([1.0 if meaning(utt, obj) else 0.0 for obj in objects])27 probs = weights / weights.sum()28 return probs # distribution over objects (indices)2930# Speaker: P(utt | obj) proportional to exp(alpha * log P_L0(obj | utt))31def speaker(obj_idx):32 obj = objects[obj_idx]33 log_weights = []34 for utt in utterances:35 ll_probs = literal_listener(utt)36 log_p = torch.log(ll_probs[obj_idx] + 1e-30)37 log_weights.append(alpha * log_p)38 log_weights = torch.tensor(log_weights)39 # Subtract max for numerical stability40 log_weights = log_weights - log_weights.max()41 weights = log_weights.exp()42 probs = weights / weights.sum()43 return probs # distribution over utterances4445# Pragmatic listener L1: P(obj | utt) proportional to speaker(obj) * uniform prior46def pragmatic_listener(utt):47 utt_idx = utterances.index(utt)48 weights = []49 for obj_idx in range(len(objects)):50 sp_probs = speaker(obj_idx)51 weights.append(sp_probs[utt_idx])52 weights = torch.stack(weights)53 probs = weights / weights.sum()54 return probs # distribution over objects5556# Speaker2: P(utt | obj) proportional to exp(alpha * log P_L1(obj | utt))57def speaker2(obj_idx):58 log_weights = []59 for utt in utterances:60 pl_probs = pragmatic_listener(utt)61 log_p = torch.log(pl_probs[obj_idx] + 1e-30)62 log_weights.append(alpha * log_p)63 log_weights = torch.tensor(log_weights)64 log_weights = log_weights - log_weights.max()65 weights = log_weights.exp()66 probs = weights / weights.sum()67 return probs # distribution over utterances6869# Listener3 L2: P(obj | utt) proportional to speaker2(obj) * uniform prior70def listener3(utt):71 utt_idx = utterances.index(utt)72 weights = []73 for obj_idx in range(len(objects)):74 sp2_probs = speaker2(obj_idx)75 weights.append(sp2_probs[utt_idx])76 weights = torch.stack(weights)77 probs = weights / weights.sum()78 return probs # distribution over objects7980# Compute L1 and L2 for 'blue'81l1_probs = pragmatic_listener('blue')82l2_probs = listener3('blue')8384# Sort objects canonically (alphabetical by JSON serialization)85import json86def obj_key(obj):87 return json.dumps(obj, sort_keys=True)8889sorted_indices = sorted(range(len(objects)), key=lambda i: obj_key(objects[i]))90sorted_objects = [objects[i] for i in sorted_indices]91l1_sorted = [l1_probs[i].item() for i in sorted_indices]92l2_sorted = [l2_probs[i].item() for i in sorted_indices]9394ANSWER = {95 'L1': {96 '__kind': 'distribution',97 'probs': l1_sorted,98 'support': sorted_objects99 },100 'L2': {101 '__kind': 'distribution',102 'probs': l2_sorted,103 'support': sorted_objects104 }105}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 1.8e-8 | 0.0000 | · |
You are an epidemiologist modeling causes of death. There are two diseases: cancer (rare, P=0.00001, often fatal P(death|cancer)=0.9) and the common cold (P=0.2, rarely fatal P(death|cold)=0.00006). Very rarely people die of other causes (P=0.000000001). death = death_by_cancer OR death_by_cold OR other_death.
Using Pyro with pyro, pyro.distributions as dist, and torch pre-imported, compute four posterior distributions over cancer (support: [false, true] represented as booleans) by manual enumeration over all discrete latent combinations:
- prior: unconditional marginal over cancer
- death: posterior given death==True
- deathAndCold: posterior given death==True and cold==True
- deathAndNoCold: posterior given death==True and cold==False
Bind ANSWER to a dict with keys 'prior', 'death', 'deathAndCold', 'deathAndNoCold', each being a dict with __kind='distribution', probs=[P(cancer=False), P(cancer=True)], and support=[False, True].
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1import itertools23p_cancer = 0.000014p_cold = 0.25p_dbc_given_cancer = 0.96p_dbcold_given_cold = 0.000067p_other = 0.00000000189# Enumerate: cancer, cold, death_by_cancer, death_by_cold, other_death10# Each is bool1112def compute_posterior(condition_fn):13 total = [0.0, 0.0] # index 0: cancer=False, index 1: cancer=True14 for cancer in [False, True]:15 for cold in [False, True]:16 for dbc in [False, True]:17 for dbcold in [False, True]:18 for od in [False, True]:19 # death_by_cancer only possible if cancer20 if dbc and not cancer:21 continue22 # death_by_cold only possible if cold23 if dbcold and not cold:24 continue2526 death = dbc or dbcold or od2728 # compute joint probability29 p = 1.030 p *= p_cancer if cancer else (1 - p_cancer)31 p *= p_cold if cold else (1 - p_cold)32 if cancer:33 p *= p_dbc_given_cancer if dbc else (1 - p_dbc_given_cancer)34 else:35 p *= 0.0 if dbc else 1.036 if cold:37 p *= p_dbcold_given_cold if dbcold else (1 - p_dbcold_given_cold)38 else:39 p *= 0.0 if dbcold else 1.040 p *= p_other if od else (1 - p_other)4142 if condition_fn(cancer, cold, death):43 total[1 if cancer else 0] += p4445 s = total[0] + total[1]46 return {"__kind": "distribution", "probs": [total[0]/s, total[1]/s], "support": [False, True]}4748prior_dist = compute_posterior(lambda cancer, cold, death: True)49death_dist = compute_posterior(lambda cancer, cold, death: death)50death_and_cold_dist = compute_posterior(lambda cancer, cold, death: death and cold)51death_and_no_cold_dist = compute_posterior(lambda cancer, cold, death: death and not cold)5253ANSWER = {54 "prior": prior_dist,55 "death": death_dist,56 "deathAndCold": death_and_cold_dist,57 "deathAndNoCold": death_and_no_cold_dist58}59
1import itertools23# Parameters4p_cancer = 0.000015p_cold = 0.26p_death_given_cancer = 0.97p_death_given_cold = 0.000068p_other_death = 0.000000001910def compute_posterior(observe_death=None, observe_cold=None):11 """12 Enumerate over (cancer, cold) combinations.13 observe_death: None or bool14 observe_cold: None or bool15 Returns [P(cancer=False), P(cancer=True)]16 """17 log_weights = {}1819 for cancer in [False, True]:20 for cold in [False, True]:21 # Prior probabilities22 p_c = p_cancer if cancer else (1 - p_cancer)23 p_co = p_cold if cold else (1 - p_cold)2425 # Death probability26 death_by_cancer = p_death_given_cancer if cancer else 0.027 death_by_cold = p_death_given_cold if cold else 0.028 other = p_other_death2930 # P(death) = 1 - P(no death) = 1 - (1-death_by_cancer)*(1-death_by_cold)*(1-other)31 p_death = 1.0 - (1.0 - death_by_cancer) * (1.0 - death_by_cold) * (1.0 - other)32 p_no_death = 1.0 - p_death3334 weight = p_c * p_co3536 if observe_death is not None:37 if observe_death:38 weight *= p_death39 else:40 weight *= p_no_death4142 if observe_cold is not None:43 if observe_cold:44 weight *= p_cold if cold else 0.045 # Actually cold is already factored in p_co, but we need to condition on cold46 # Let me redo: the cold observation is already in the enumeration47 # We just skip combinations that don't match48 pass4950 key = (cancer, cold)51 log_weights[key] = weight5253 # Apply cold conditioning by zeroing out non-matching54 if observe_cold is not None:55 for key in list(log_weights.keys()):56 cancer, cold = key57 if cold != observe_cold:58 log_weights[key] = 0.05960 # Marginalize over cold to get P(cancer)61 p_cancer_false = sum(w for (c, co), w in log_weights.items() if not c)62 p_cancer_true = sum(w for (c, co), w in log_weights.items() if c)6364 total = p_cancer_false + p_cancer_true65 p_cancer_false /= total66 p_cancer_true /= total6768 return [p_cancer_false, p_cancer_true]6970# Compute the four posteriors71prior_probs = compute_posterior(observe_death=None, observe_cold=None)72death_probs = compute_posterior(observe_death=True, observe_cold=None)73death_and_cold_probs = compute_posterior(observe_death=True, observe_cold=True)74death_and_no_cold_probs = compute_posterior(observe_death=True, observe_cold=False)7576def make_dist_dict(probs):77 return {78 '__kind': 'distribution',79 'probs': probs,80 'support': [False, True]81 }8283ANSWER = {84 'prior': make_dist_dict(prior_probs),85 'death': make_dist_dict(death_probs),86 'deathAndCold': make_dist_dict(death_and_cold_probs),87 'deathAndNoCold': make_dist_dict(death_and_no_cold_probs),88}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 3.1e-12 | 0.0000 | · |
Imagine that you are an epidemiologist determining people's cause of death. There are two main diseases: cancer (rare, P = 0.00001, often fatal P(death|cancer)=0.9) and the common cold (P = 0.2, rarely fatal P(death|cold)=0.00006). Very rarely, people die of other causes (P = 0.000000001).
Using Pyro (with pyro, pyro.distributions as dist, and torch pre-imported), compute four posterior distributions over the boolean variable cold by manually enumerating the discrete latent space:
- prior: unconditional distribution over cold
- death: distribution over cold given death == True
- deathAndCancer: distribution over cold given death == True and cancer == True
- deathAndNoCancer: distribution over cold given death == True and cancer == False
For each marginal, enumerate all combinations of (cancer, cold, death_by_cancer, death_by_cold, other_death), compute the joint probability, filter by the condition, and marginalize over cold. Represent each as a dict with __kind='distribution', probs=[P(cold=False), P(cold=True)], and support=[False, True].
Bind the result to a top-level variable ANSWER as a dict with keys prior, death, deathAndCancer, deathAndNoCancer.
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WebPPL primer
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1import itertools23p_cancer = 0.000014p_cold = 0.25p_death_given_cancer = 0.96p_death_given_cold = 0.000067p_other_death = 0.00000000189# Enumerate over (cancer, cold, death_by_cancer, death_by_cold, other_death)10# Each is boolean (False/True)1112def compute_marginal(condition_fn):13 prob_cold_false = 0.014 prob_cold_true = 0.015 for cancer in [False, True]:16 p_c = p_cancer if cancer else (1 - p_cancer)17 for cold in [False, True]:18 p_co = p_cold if cold else (1 - p_cold)19 for dbc in [False, True]:20 if cancer:21 p_dbc = p_death_given_cancer if dbc else (1 - p_death_given_cancer)22 else:23 p_dbc = 0.0 if dbc else 1.024 for dbco in [False, True]:25 if cold:26 p_dbco = p_death_given_cold if dbco else (1 - p_death_given_cold)27 else:28 p_dbco = 0.0 if dbco else 1.029 for od in [False, True]:30 p_od = p_other_death if od else (1 - p_other_death)31 death = dbc or dbco or od32 joint = p_c * p_co * p_dbc * p_dbco * p_od33 if condition_fn(cancer, cold, death):34 if cold:35 prob_cold_true += joint36 else:37 prob_cold_false += joint38 total = prob_cold_false + prob_cold_true39 if total == 0:40 return {"__kind": "distribution", "probs": [0.5, 0.5], "support": [False, True]}41 return {42 "__kind": "distribution",43 "probs": [prob_cold_false / total, prob_cold_true / total],44 "support": [False, True]45 }4647prior = compute_marginal(lambda cancer, cold, death: True)48death_marginal = compute_marginal(lambda cancer, cold, death: death)49death_and_cancer = compute_marginal(lambda cancer, cold, death: death and cancer)50death_and_no_cancer = compute_marginal(lambda cancer, cold, death: death and not cancer)5152ANSWER = {53 "prior": prior,54 "death": death_marginal,55 "deathAndCancer": death_and_cancer,56 "deathAndNoCancer": death_and_no_cancer57}58
1import itertools23# Parameters4p_cancer = 0.000015p_cold = 0.26p_death_given_cancer = 0.97p_death_given_cold = 0.000068p_other_death = 0.000000001910def compute_joint(cancer, cold, death_by_cancer, death_by_cold, other_death):11 """Compute joint probability of all variables."""12 # P(cancer)13 p = p_cancer if cancer else (1 - p_cancer)14 # P(cold)15 p *= p_cold if cold else (1 - p_cold)16 # P(death_by_cancer | cancer)17 if cancer:18 p *= p_death_given_cancer if death_by_cancer else (1 - p_death_given_cancer)19 else:20 p *= 0.0 if death_by_cancer else 1.021 # P(death_by_cold | cold)22 if cold:23 p *= p_death_given_cold if death_by_cold else (1 - p_death_given_cold)24 else:25 p *= 0.0 if death_by_cold else 1.026 # P(other_death)27 p *= p_other_death if other_death else (1 - p_other_death)28 return p2930# Enumerate all combinations31bool_vals = [False, True]32all_combos = list(itertools.product(bool_vals, bool_vals, bool_vals, bool_vals, bool_vals))3334def make_distribution(condition_fn):35 """Compute posterior over cold given condition."""36 prob_cold_false = 0.037 prob_cold_true = 0.03839 for (cancer, cold, death_by_cancer, death_by_cold, other_death) in all_combos:40 death = death_by_cancer or death_by_cold or other_death4142 if not condition_fn(cancer, cold, death, death_by_cancer, death_by_cold, other_death):43 continue4445 p = compute_joint(cancer, cold, death_by_cancer, death_by_cold, other_death)4647 if cold:48 prob_cold_true += p49 else:50 prob_cold_false += p5152 total = prob_cold_false + prob_cold_true53 if total > 0:54 prob_cold_false /= total55 prob_cold_true /= total5657 return {58 '__kind': 'distribution',59 'probs': [prob_cold_false, prob_cold_true],60 'support': [False, True]61 }6263# Prior: unconditional64prior = make_distribution(lambda cancer, cold, death, dbc, dbd, od: True)6566# Death: given death == True67death_dist = make_distribution(lambda cancer, cold, death, dbc, dbd, od: death == True)6869# DeathAndCancer: given death == True and cancer == True70death_and_cancer = make_distribution(lambda cancer, cold, death, dbc, dbd, od: death == True and cancer == True)7172# DeathAndNoCancer: given death == True and cancer == False73death_and_no_cancer = make_distribution(lambda cancer, cold, death, dbc, dbd, od: death == True and cancer == False)7475ANSWER = {76 'prior': prior,77 'death': death_dist,78 'deathAndCancer': death_and_cancer,79 'deathAndNoCancer': death_and_no_cancer80}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
I flip a fair coin. Compute the probability that it lands heads. End your program with the numeric answer bound to a top-level variable named ANSWER. Note that pyro, pyro.distributions as dist, and torch are pre-imported.
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1ANSWER = 0.5
1ANSWER = 0.5
0.5000
0.5000
| run | flags | bucket | TV | KL | actions |
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ✓val+ | — | — | · |
I have a biased coin with P(heads) = 0.9, and a fair coin. Someone hands me one of the two coins, chosen uniformly at random, without telling me which. I flip it three times. Given that the first two flips landed heads, what is the posterior distribution over the third flip?
Return as a dist.Categorical(probs=...) over support [0, 1] (0=tails, 1=heads). End your program with ANSWER = dist.Categorical(probs=torch.tensor([p_tails, p_heads])).
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1import itertools2import torch34result = {0.0: 0.0, 1.0: 0.0}5for coin_fair, f1, f2, f3 in itertools.product([0, 1], repeat=4):6 p_per_flip = 0.5 if coin_fair == 1 else 0.97 pf = lambda v: p_per_flip if v == 1 else (1 - p_per_flip)8 joint = 0.5 * pf(f1) * pf(f2) * pf(f3)9 if f1 == 1 and f2 == 1:10 result[float(f3)] += joint1112total = sum(result.values())13probs = torch.tensor([result[0.0] / total, result[1.0] / total])14ANSWER = dist.Categorical(probs=probs)15
1import torch2import pyro3import pyro.distributions as dist45# Prior probabilities6p_A = 0.57p_B = 0.589# Likelihoods of observing HH10p_HH_given_A = 0.9 ** 211p_HH_given_B = 0.5 ** 21213# Unnormalized posterior over coins14unnorm_A = p_HH_given_A * p_A15unnorm_B = p_HH_given_B * p_B1617total = unnorm_A + unnorm_B1819post_A = unnorm_A / total20post_B = unnorm_B / total2122# Posterior predictive for third flip23p_heads = post_A * 0.9 + post_B * 0.524p_tails = post_A * 0.1 + post_B * 0.52526p_heads_tensor = torch.tensor(p_heads)27p_tails_tensor = torch.tensor(p_tails)2829ANSWER = dist.Categorical(probs=torch.tensor([p_tails, p_heads]))
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
I have a biased coin with P(heads) = 0.9, and a fair coin. Someone hands me one of the two coins, chosen uniformly at random, without telling me which. I flip it three times. Given that all three flips landed heads, what is the posterior distribution over the coin type?
Return as a distribution over the string labels 'biased' and 'fair'. Since Pyro's dist.Categorical requires integer support, build the answer as a dict literal matching the cross-PPL schema: ANSWER = {"__kind": "distribution", "probs": [p_biased, p_fair], "support": ['biased', 'fair']}. (Labels must be sorted; 'biased' < 'fair'.)
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1import itertools2import torch34result = {'fair': 0.0, 'biased': 0.0}5for coin_fair, f1, f2, f3 in itertools.product([0, 1], repeat=4):6 p_per_flip = 0.5 if coin_fair == 1 else 0.97 pf = lambda v: p_per_flip if v == 1 else (1 - p_per_flip)8 joint = 0.5 * pf(f1) * pf(f2) * pf(f3)9 if f1 == 1 and f2 == 1 and f3 == 1:10 key = 'fair' if coin_fair == 1 else 'biased'11 result[key] += joint12total = sum(result.values())13labels = sorted(result.keys())14probs = [result[k] / total for k in labels]15ANSWER = {"__kind": "distribution", "probs": probs, "support": labels}16
1import pyro2import pyro.distributions as dist3import torch45# Prior: uniform over biased and fair6# P(biased) = 0.5, P(fair) = 0.578# Likelihood of 3 heads:9# P(HHH | biased) = 0.9^310# P(HHH | fair) = 0.5^31112p_biased_prior = 0.513p_fair_prior = 0.51415p_hhh_given_biased = 0.9 ** 316p_hhh_given_fair = 0.5 ** 31718# Unnormalized posteriors19unnorm_biased = p_biased_prior * p_hhh_given_biased20unnorm_fair = p_fair_prior * p_hhh_given_fair2122# Normalize23total = unnorm_biased + unnorm_fair24p_biased_post = unnorm_biased / total25p_fair_post = unnorm_fair / total2627ANSWER = {28 "__kind": "distribution",29 "probs": [p_biased_post, p_fair_post],30 "support": ["biased", "fair"]31}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
I have a biased coin with P(heads) = 0.9, and a fair coin. Someone hands me one of the two coins, chosen uniformly at random, without telling me which. I flip it three times. Given that the first two flips were different (one heads, one tails, in either order), what is the posterior distribution over the third flip?
Return as a dist.Categorical(probs=...) over support [0, 1] (0=tails, 1=heads). End your program with ANSWER = dist.Categorical(probs=torch.tensor([p_tails, p_heads])).
system prompt
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WebPPL primer
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1import itertools2import torch34result = {0.0: 0.0, 1.0: 0.0}5for coin_fair, f1, f2, f3 in itertools.product([0, 1], repeat=4):6 p_per_flip = 0.5 if coin_fair == 1 else 0.97 pf = lambda v: p_per_flip if v == 1 else (1 - p_per_flip)8 joint = 0.5 * pf(f1) * pf(f2) * pf(f3)9 if f1 != f2:10 result[float(f3)] += joint1112total = sum(result.values())13probs = torch.tensor([result[0.0] / total, result[1.0] / total])14ANSWER = dist.Categorical(probs=probs)15
1import pyro2import pyro.distributions as dist3import torch45# Two coins: fair (p=0.5) and biased (p=0.9)6p_fair = 0.57p_biased = 0.989# Prior over coins10prior = torch.tensor([0.5, 0.5]) # [fair, biased]11p_heads = torch.tensor([p_fair, p_biased])1213# P(first two flips are different | coin)14# = P(HT) + P(TH) = p*(1-p) + (1-p)*p = 2*p*(1-p)15p_diff_given_coin = 2 * p_heads * (1 - p_heads)16# fair: 2*0.5*0.5 = 0.517# biased: 2*0.9*0.1 = 0.181819# Joint: prior * likelihood20joint = prior * p_diff_given_coin2122# Posterior over coin23posterior_coin = joint / joint.sum()2425# P(third flip = H | diff) = sum over coins of P(coin | diff) * P(H | coin)26p_heads_third = (posterior_coin * p_heads).sum()27p_tails_third = 1 - p_heads_third2829ANSWER = dist.Categorical(probs=torch.tensor([p_tails_third, p_heads_third]))
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 6.0e-8 | 0.0000 | · |
Using Pyro (with pyro, pyro.distributions as dist, and torch pre-imported), implement the following Bayesian model and bind the result to a top-level variable ANSWER.
Model description:
- Alice has a stable 'nice' trait: nice ~ Bernoulli(0.7)
- Whether Alice wants something today: wants ~ Bernoulli(0.2 if nice else 0.5)
- Alice smiles if EITHER of two independent Bernoulli flips is true:
- If she wants something: flip(0.8), otherwise flip(0.5)
- If she is nice: flip(0.8), otherwise flip(0.5)
(i.e., smiles = flip1 OR flip2)
Compute the marginal distribution of whether Alice smiles today by exact enumeration over (nice, wants). The result should be ANSWER = {"__kind": "distribution", "probs": [P(smiles=False), P(smiles=True)], "support": [false, true]} (use Python booleans False/True as support).
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1p_nice = 0.723# Enumerate over (nice, wants)4combinations = [5 (True, True),6 (True, False),7 (False, True),8 (False, False),9]1011p_smiles_true = 0.01213for nice, wants in combinations:14 # Prior probability of this combination15 p_n = p_nice if nice else (1 - p_nice)16 p_w = (0.2 if nice else 0.5) if wants else (1 - (0.2 if nice else 0.5))17 weight = p_n * p_w1819 # P(flip1) based on wants20 p_flip1 = 0.8 if wants else 0.521 # P(flip2) based on nice22 p_flip2 = 0.8 if nice else 0.52324 # P(smiles) = P(flip1 OR flip2) = 1 - P(not flip1) * P(not flip2)25 p_smiles = 1.0 - (1.0 - p_flip1) * (1.0 - p_flip2)2627 p_smiles_true += weight * p_smiles2829p_smiles_false = 1.0 - p_smiles_true3031ANSWER = {"__kind": "distribution", "probs": [p_smiles_false, p_smiles_true], "support": [False, True]}32
1import itertools23# Enumerate over (nice, wants)4p_smiles_false = 0.05p_smiles_true = 0.067for nice in [0, 1]:8 p_nice = 0.7 if nice == 1 else 0.3910 for wants in [0, 1]:11 p_wants_given_nice = 0.2 if nice == 1 else 0.512 p_wants = p_wants_given_nice if wants == 1 else (1.0 - p_wants_given_nice)1314 # flip1: based on wants15 p_flip1 = 0.8 if wants == 1 else 0.516 # flip2: based on nice17 p_flip2 = 0.8 if nice == 1 else 0.51819 # smiles = flip1 OR flip220 p_smiles_given = 1.0 - (1.0 - p_flip1) * (1.0 - p_flip2)2122 joint = p_nice * p_wants23 p_smiles_true += joint * p_smiles_given24 p_smiles_false += joint * (1.0 - p_smiles_given)2526total = p_smiles_false + p_smiles_true27p_smiles_false /= total28p_smiles_true /= total2930ANSWER = {"__kind": "distribution", "probs": [p_smiles_false, p_smiles_true], "support": [False, True]}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Given the following extended smiles model semantics in Pyro:
- Bob's niceness is stable (memoized): nice_bob ~ Bernoulli(0.7)
- Each day, whether Bob wants something: wants ~ Bernoulli(0.2 if nice else 0.5) (independent each day)
- smiles(wants, nice) = logical OR of two independent flips: Bernoulli(0.8 if wants else 0.5) OR Bernoulli(0.8 if nice else 0.5)
Observations: Bob was NOT smiling on 5 separate past days (each with an independent wantsSomething draw). Today, he IS smiling (using wants_today, an independent draw for today).
Using exact enumeration over all discrete latents (nice_bob, wants_today, and 5 independent past wants variables), compute the posterior distribution over wants_today. Use pyro, pyro.distributions as dist, and torch (pre-imported). Bind the result to a top-level variable ANSWER as a {"__kind": "distribution", "probs": [...], "support": [...]} dict with support [false, true] (use Python booleans False, True).
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1import itertools2import math34# nice_bob: bool5# wants_today: bool6# wants_past[i]: bool for i in 0..478def p_smiles(wants, nice):9 pw = 0.8 if wants else 0.510 pn = 0.8 if nice else 0.511 return 1.0 - (1.0 - pw) * (1.0 - pn)1213def p_not_smiles(wants, nice):14 return 1.0 - p_smiles(wants, nice)1516log_weights = {False: 0.0, True: 0.0}1718for nice_bob in [False, True]:19 p_nice = 0.7 if nice_bob else 0.320 p_wants_given_nice = 0.2 if nice_bob else 0.521 p_not_wants_given_nice = 1.0 - p_wants_given_nice2223 # P(not smiles on one past day | nice_bob)24 # marginalize over wants_past25 p_not_smiles_one_day = (26 p_not_wants_given_nice * p_not_smiles(False, nice_bob) +27 p_wants_given_nice * p_not_smiles(True, nice_bob)28 )29 # 5 past days all not smiling30 p_past = p_not_smiles_one_day ** 53132 for wants_today in [False, True]:33 p_wants_today = p_wants_given_nice if wants_today else (1.0 - p_wants_given_nice)34 p_smiles_today = p_smiles(wants_today, nice_bob)3536 weight = p_nice * p_past * p_wants_today * p_smiles_today37 log_weights[wants_today] += weight3839total = log_weights[False] + log_weights[True]40prob_false = log_weights[False] / total41prob_true = log_weights[True] / total4243ANSWER = {"__kind": "distribution", "probs": [prob_false, prob_true], "support": [False, True]}44
1import itertools23# Bob's niceness: nice_bob in {0, 1}4# wants_today in {0, 1}5# 5 past wants variables, each in {0, 1}67# Prior: nice_bob ~ Bernoulli(0.7)8# wants ~ Bernoulli(0.2 if nice else 0.5) each day independently9# smiles(wants, nice) = Bernoulli(0.8 if wants else 0.5) OR Bernoulli(0.8 if nice else 0.5)10# P(smiles=1 | wants, nice) = 1 - P(flip1=0)*P(flip2=0)11# = 1 - (1 - (0.8 if wants else 0.5)) * (1 - (0.8 if nice else 0.5))1213def p_smiles(wants, nice):14 p1 = 0.8 if wants else 0.515 p2 = 0.8 if nice else 0.516 return 1.0 - (1.0 - p1) * (1.0 - p2)1718def p_wants(nice):19 return 0.2 if nice else 0.52021import torch2223# Enumerate over nice_bob, wants_today, and 5 past wants24# Total: 2 * 2 * 2^5 = 128 combinations2526log_weights_wants0 = []27log_weights_wants1 = []2829for nice in [0, 1]:30 # P(nice)31 log_p_nice = torch.log(torch.tensor(0.7 if nice == 1 else 0.3))3233 for wants_today in [0, 1]:34 # P(wants_today | nice)35 pw = p_wants(nice)36 log_p_wants_today = torch.log(torch.tensor(pw if wants_today == 1 else 1.0 - pw))3738 # P(smiles_today=1 | wants_today, nice)39 ps_today = p_smiles(wants_today, nice)40 log_p_obs_today = torch.log(torch.tensor(ps_today))4142 # Sum over 5 past days43 # Each past day: wants_i ~ Bernoulli(p_wants(nice)), smiles_i=044 # P(smiles_i=0 | wants_i, nice) = 1 - p_smiles(wants_i, nice)4546 log_past_sum = torch.tensor(0.0)47 # For each past day, marginalize over wants_i48 for day in range(5):49 # Marginalize wants_i50 log_day_marginal = torch.tensor(float('-inf'))51 for wants_i in [0, 1]:52 pw_i = p_wants(nice)53 log_p_wi = torch.log(torch.tensor(pw_i if wants_i == 1 else 1.0 - pw_i))54 ps_i = p_smiles(wants_i, nice)55 log_p_not_smile_i = torch.log(torch.tensor(1.0 - ps_i))56 log_contrib = log_p_wi + log_p_not_smile_i57 log_day_marginal = torch.logaddexp(log_day_marginal, log_contrib)58 log_past_sum = log_past_sum + log_day_marginal5960 log_w = log_p_nice + log_p_wants_today + log_p_obs_today + log_past_sum6162 if wants_today == 0:63 log_weights_wants0.append(log_w)64 else:65 log_weights_wants1.append(log_w)6667# Combine log weights for each value of wants_today68def logsumexp_list(lst):69 t = torch.stack(lst)70 return torch.logsumexp(t, dim=0)7172log_w0 = logsumexp_list(log_weights_wants0)73log_w1 = logsumexp_list(log_weights_wants1)7475# Normalize76log_total = torch.logaddexp(log_w0, log_w1)77p_wants_today_false = torch.exp(log_w0 - log_total).item()78p_wants_today_true = torch.exp(log_w1 - log_total).item()7980ANSWER = {81 "__kind": "distribution",82 "probs": [p_wants_today_false, p_wants_today_true],83 "support": [False, True]84}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 9.4e-9 | 0.0000 | · |
I have a sprinkler in my garden that turns on each morning at random - half the time, independently each day. I live in a city where it rains on 30% of mornings. The lawn gets wet whenever the sprinkler turns on, it rains, or both. One morning, I notice my lawn is wet. Using Pyro (with pyro, pyro.distributions as dist, and torch pre-imported), compute the exact posterior distributions by manual enumeration over the discrete latent space. Return a dict named ANSWER with two keys:
- rain: posterior distribution over whether it rained (as a {"__kind": "distribution", "probs": [...], "support": [...]} dict with boolean support [False, True] sorted canonically)
- sprinkler: posterior distribution over whether the sprinkler turned on (same format)
The support for each should be sorted as [False, True] and probs should correspond to those support values.
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1# Enumerate over (sprinkler, rain) in {True, False}^22# P(sprinkler=True) = 0.5, P(rain=True) = 0.33# condition on wetLawn = sprinkler or rain45import itertools67p_sprinkler = 0.58p_rain = 0.3910# Compute joint unnormalized weights conditioned on wetLawn11# For rain posterior12rain_false_weight = 0.013rain_true_weight = 0.01415for s in [False, True]:16 for r in [False, True]:17 wet = s or r18 if not wet:19 continue20 ps = p_sprinkler if s else (1 - p_sprinkler)21 pr = p_rain if r else (1 - p_rain)22 w = ps * pr23 if r:24 rain_true_weight += w25 else:26 rain_false_weight += w2728rain_total = rain_false_weight + rain_true_weight29rain_prob_false = rain_false_weight / rain_total30rain_prob_true = rain_true_weight / rain_total3132# For sprinkler posterior33sprinkler_false_weight = 0.034sprinkler_true_weight = 0.03536for s in [False, True]:37 for r in [False, True]:38 wet = s or r39 if not wet:40 continue41 ps = p_sprinkler if s else (1 - p_sprinkler)42 pr = p_rain if r else (1 - p_rain)43 w = ps * pr44 if s:45 sprinkler_true_weight += w46 else:47 sprinkler_false_weight += w4849sprinkler_total = sprinkler_false_weight + sprinkler_true_weight50sprinkler_prob_false = sprinkler_false_weight / sprinkler_total51sprinkler_prob_true = sprinkler_true_weight / sprinkler_total5253ANSWER = {54 "rain": {55 "__kind": "distribution",56 "probs": [rain_prob_false, rain_prob_true],57 "support": [False, True]58 },59 "sprinkler": {60 "__kind": "distribution",61 "probs": [sprinkler_prob_false, sprinkler_prob_true],62 "support": [False, True]63 }64}65
1import itertools23# Prior probabilities4p_rain = 0.35p_sprinkler = 0.567# Enumerate all combinations of (rain, sprinkler)8# rain in {0, 1}, sprinkler in {0, 1}9log_weights = {}10for rain in [0, 1]:11 for sprinkler in [0, 1]:12 # Prior log prob13 lp_rain = torch.tensor(rain * p_rain + (1 - rain) * (1 - p_rain)).log()14 lp_sprinkler = torch.tensor(sprinkler * p_sprinkler + (1 - sprinkler) * (1 - p_sprinkler)).log()1516 # Likelihood: lawn is wet if rain or sprinkler17 wet = (rain == 1) or (sprinkler == 1)18 lp_obs = torch.tensor(0.0) if wet else torch.tensor(float('-inf'))1920 log_weights[(rain, sprinkler)] = lp_rain + lp_sprinkler + lp_obs2122# Convert to weights23keys = list(log_weights.keys())24log_w = torch.stack([log_weights[k] for k in keys])25w = log_w.exp()26w = w / w.sum()2728# Compute marginal for rain29# P(rain=0 | wet), P(rain=1 | wet)30p_rain_false = sum(w[i] for i, (r, s) in enumerate(keys) if r == 0)31p_rain_true = sum(w[i] for i, (r, s) in enumerate(keys) if r == 1)3233# Compute marginal for sprinkler34p_sprinkler_false = sum(w[i] for i, (r, s) in enumerate(keys) if s == 0)35p_sprinkler_true = sum(w[i] for i, (r, s) in enumerate(keys) if s == 1)3637rain_probs = [p_rain_false.item(), p_rain_true.item()]38sprinkler_probs = [p_sprinkler_false.item(), p_sprinkler_true.item()]3940ANSWER = {41 "rain": {42 "__kind": "distribution",43 "probs": rain_probs,44 "support": [False, True]45 },46 "sprinkler": {47 "__kind": "distribution",48 "probs": sprinkler_probs,49 "support": [False, True]50 }51}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 4.6e-9 | -0.0000 | · |
Model a garden sprinkler scenario in Pyro. The sprinkler turns on each morning with probability 0.5 (independently each day). It rains on 30% of mornings. The lawn gets wet whenever the sprinkler is on, it rains, or both. A neighbor Kelsey has the same kind of sprinkler (independent, also probability 0.5). One morning, both my lawn and Kelsey's lawn are wet. Compute the exact posterior distribution over whether it rained using manual enumeration over the discrete latent space. pyro, pyro.distributions as dist, and torch are pre-imported. End your program with a top-level assignment ANSWER = {"__kind": "distribution", "probs": [...], "support": [false, true]} where the probabilities correspond to rain=False and rain=True respectively.
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1import itertools23# Enumerate over all combinations of (rain, mySprinkler, herSprinkler)4# each is boolean (0 or 1)5total_weight = 0.06rain_weight = 0.078for rain in [False, True]:9 for my_sprinkler in [False, True]:10 for her_sprinkler in [False, True]:11 p_rain = 0.3 if rain else 0.712 p_my = 0.513 p_her = 0.514 p = p_rain * p_my * p_her15 my_wet = my_sprinkler or rain16 her_wet = her_sprinkler or rain17 if my_wet and her_wet:18 total_weight += p19 if rain:20 rain_weight += p2122p_rain_true = rain_weight / total_weight23p_rain_false = 1.0 - p_rain_true2425ANSWER = {"__kind": "distribution", "probs": [p_rain_false, p_rain_true], "support": [False, True]}26
1import itertools23# Latent variables: rain (0/1), my_sprinkler (0/1), kelsey_sprinkler (0/1)4# Observations: my_lawn_wet = True, kelsey_lawn_wet = True56p_rain = 0.37p_sprinkler = 0.589log_weights = {0: 0.0, 1: 0.0}1011for rain in [0, 1]:12 for my_s in [0, 1]:13 for k_s in [0, 1]:14 # Prior probabilities15 p_r = p_rain if rain == 1 else (1 - p_rain)16 p_ms = p_sprinkler if my_s == 1 else (1 - p_sprinkler)17 p_ks = p_sprinkler if k_s == 1 else (1 - p_sprinkler)1819 # Lawn wet = sprinkler OR rain20 my_wet = (my_s == 1) or (rain == 1)21 kelsey_wet = (k_s == 1) or (rain == 1)2223 # Both lawns must be wet (observation)24 if my_wet and kelsey_wet:25 log_weights[rain] += p_r * p_ms * p_ks2627# Normalize28total = log_weights[0] + log_weights[1]29p_no_rain = log_weights[0] / total30p_rain_post = log_weights[1] / total3132ANSWER = {"__kind": "distribution", "probs": [p_no_rain, p_rain_post], "support": [False, True]}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 2.8e-17 | -0.0000 | · |
Model a garden sprinkler scenario in Pyro. There is a sprinkler that turns on each morning with probability 0.5, independently for each person. It rains on 30% of mornings. A lawn is wet whenever the sprinkler turns on, it rains, or both. There are five people: me, Kelsey, Kevin, Manu, and Josh — each with an independent sprinkler (p=0.5). One morning, all five lawns are wet. Compute the exact posterior over whether it rained using enumeration. pyro, pyro.distributions as dist, and torch are pre-imported. Bind the result to a top-level variable ANSWER as a dict with __kind, probs, and support keys, where support is [false, true] (Python False/True).
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1import itertools23# Enumerate all combinations of (rain, s_me, s_kelsey, s_kevin, s_manu, s_josh)4# Each is boolean (0 or 1)56total_weight = 0.07rain_true_weight = 0.089for rain in [0, 1]:10 p_rain = 0.3 if rain else 0.711 for sprinklers in itertools.product([0, 1], repeat=5):12 p_sprinklers = 1.013 for s in sprinklers:14 p_sprinklers *= 0.515 # All lawns must be wet16 all_wet = all((rain or s) for s in sprinklers)17 if all_wet:18 w = p_rain * p_sprinklers19 total_weight += w20 if rain:21 rain_true_weight += w2223p_rain_false = (total_weight - rain_true_weight) / total_weight24p_rain_true = rain_true_weight / total_weight2526ANSWER = {"__kind": "distribution", "probs": [p_rain_false, p_rain_true], "support": [False, True]}27
1import itertools23# Prior: P(rain) = 0.34p_rain = 0.35p_sprinkler = 0.567# Five people: me, Kelsey, Kevin, Manu, Josh8# Each has independent sprinkler with p=0.59# Lawn is wet if sprinkler OR rain1011# We enumerate over rain in {0, 1}12# For each value of rain, we compute the likelihood that all 5 lawns are wet1314log_weights = []1516for rain in [0, 1]:17 # Prior log prob of rain18 if rain == 1:19 log_prior = torch.log(torch.tensor(p_rain))20 else:21 log_prior = torch.log(torch.tensor(1 - p_rain))2223 # For each person, P(wet | rain) = P(sprinkler=1 OR rain=1)24 # = 1 - P(sprinkler=0 AND rain=0)25 # If rain=1: P(wet) = 1 (since rain alone makes it wet)26 # If rain=0: P(wet) = P(sprinkler=1) = 0.52728 if rain == 1:29 p_wet = 1.030 else:31 p_wet = p_sprinkler # 0.53233 # All 5 lawns are wet34 log_likelihood = 5 * torch.log(torch.tensor(p_wet))3536 log_weights.append(log_prior + log_likelihood)3738log_weights_tensor = torch.stack(log_weights)39weights = log_weights_tensor.exp()40weights = weights / weights.sum()4142# weights[0] = P(rain=False | all wet)43# weights[1] = P(rain=True | all wet)4445ANSWER = {46 "__kind": "distribution",47 "probs": [weights[0].item(), weights[1].item()],48 "support": [False, True]49}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 5.2e-9 | 0.0000 | · |
A machine randomly draws a letter of the word "game" with probabilities {g: 0.05, a: 0.45, m: 0.05, e: 0.45}. Bob's probability of winning given letter at position k is 1/k^2 (g=1, a=2, m=3, e=4). We observe that Bob won. Compute p(letter | win) by exact enumeration over the discrete latent space.
Use pyro, pyro.distributions as dist, and torch (pre-imported). Enumerate over letters ['g', 'a', 'm', 'e'] with prior probs [0.05, 0.45, 0.05, 0.45], weight each by its win probability 1/k^2, normalize to get the posterior, and bind the result to a top-level variable ANSWER as a dict with __kind, probs, and support keys. The support must be sorted alphabetically.
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1letter_vals = ['g', 'a', 'm', 'e']2letter_probs = [0.05, 0.45, 0.05, 0.45]34# Compute unnormalized weights: prior * win_prob5weights = {}6for i, letter in enumerate(letter_vals):7 position = i + 18 win_prob = 1.0 / (position ** 2)9 weights[letter] = letter_probs[i] * win_prob1011# Normalize12total = sum(weights.values())13norm_weights = {letter: w / total for letter, w in weights.items()}1415# Sort support alphabetically16sorted_support = sorted(norm_weights.keys())17sorted_probs = [norm_weights[l] for l in sorted_support]1819ANSWER = {20 "__kind": "distribution",21 "probs": sorted_probs,22 "support": sorted_support23}24
1import torch23letters = ['a', 'e', 'g', 'm']4# positions: g=1, a=2, m=3, e=45letter_to_k = {'g': 1, 'a': 2, 'm': 3, 'e': 4}6prior_probs = {'g': 0.05, 'a': 0.45, 'm': 0.05, 'e': 0.45}78# Compute unnormalized weights: prior * likelihood9weights = []10for letter in letters: # alphabetical: a, e, g, m11 k = letter_to_k[letter]12 prior = prior_probs[letter]13 likelihood = 1.0 / (k ** 2)14 weights.append(prior * likelihood)1516w = torch.tensor(weights)17w = w / w.sum()1819ANSWER = {20 "__kind": "Categorical",21 "probs": w.tolist(),22 "support": letters23}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 5.2e-9 | -0.0000 | · |
Implement the following Bayesian model in Pyro and bind the result to a top-level variable named ANSWER.
Same casino-game setup: letters {g, a, m, e} drawn with probs {0.05, 0.45, 0.05, 0.45}, win prob = 1/k^2 with k = position in 'game' (1-indexed). Condition on Bob winning (i.e., flip with winProb is True). Return the posterior distribution over 'vowel' vs 'consonant'.
Vowels are: a, e, i, o, u. The letters and their probs: - 'g': prob 0.05, position 1, winProb = 1/1 = 1.0 - 'a': prob 0.45, position 2, winProb = 1/4 = 0.25 - 'm': prob 0.05, position 3, winProb = 1/9 - 'e': prob 0.45, position 4, winProb = 1/16
Enumerate all four letters, compute unnormalized weights = letterProb * winProb, group by 'vowel' or 'consonant', normalize, and return as ANSWER = {"__kind": "distribution", "probs": [...], "support": [...]} with support sorted alphabetically (i.e., ['consonant', 'vowel']).
pyro, pyro.distributions as dist, and torch are pre-imported.
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1letter_vals = ['g', 'a', 'm', 'e']2vowels = set(['a', 'e', 'i', 'o', 'u'])34def check_vowel(letter):5 return letter in vowels67letter_probs = [0.45 if check_vowel(l) else 0.05 for l in letter_vals]89# Enumerate: weight = letter_prob * win_prob10# win_prob = 1 / position^211weights = {}12for i, letter in enumerate(letter_vals):13 position = i + 114 win_prob = 1.0 / (position ** 2)15 category = 'vowel' if check_vowel(letter) else 'consonant'16 w = letter_probs[i] * win_prob17 weights[category] = weights.get(category, 0.0) + w1819total = sum(weights.values())2021# Support sorted alphabetically22support = sorted(weights.keys())23probs = [weights[s] / total for s in support]2425ANSWER = {"__kind": "distribution", "probs": probs, "support": support}26
1# Letter setup2letters = ['g', 'a', 'm', 'e']3letter_probs = [0.05, 0.45, 0.05, 0.45]4positions = [1, 2, 3, 4]5win_probs = [1.0 / (k**2) for k in positions]6vowels = set('aeiou')78# Compute unnormalized weights9weights = {letter: lp * wp for letter, lp, wp in zip(letters, letter_probs, win_probs)}1011# Group by vowel/consonant12consonant_weight = sum(w for l, w in weights.items() if l not in vowels)13vowel_weight = sum(w for l, w in weights.items() if l in vowels)1415total = consonant_weight + vowel_weight16consonant_prob = consonant_weight / total17vowel_prob = vowel_weight / total1819# Support sorted alphabetically: ['consonant', 'vowel']20ANSWER = {21 "__kind": "distribution",22 "probs": [consonant_prob, vowel_prob],23 "support": ["consonant", "vowel"]24}
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Write a Pyro expression that samples a boolean with probability 0.4 of being True, using a structurally different approach than a single Bernoulli(0.4). Use pyro, pyro.distributions as dist, and torch which are pre-imported. Specifically, implement the logic: flip a fair coin; if it lands heads return False, otherwise flip another coin with probability 0.8 and return its result. Generate 100 such samples and bind the list to a top-level variable named ANSWER. Each sample should be a Python bool.
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1pyro.set_rng_seed(42)23def model():4 first = pyro.sample('first', dist.Bernoulli(0.5))5 if first.item() == 1.0:6 return False7 else:8 second = pyro.sample('second', dist.Bernoulli(0.8))9 return bool(second.item() == 1.0)1011samples = []12for i in range(100):13 pyro.set_rng_seed(i + 1000)14 first = dist.Bernoulli(0.5).sample()15 if first.item() == 1.0:16 samples.append(False)17 else:18 second = dist.Bernoulli(0.8).sample()19 samples.append(bool(second.item() == 1.0))2021ANSWER = samples
1def sample_bool(i):2 fair = pyro.sample(f"fair_{i}", dist.Bernoulli(0.5))3 if fair.item() == 1.0:4 return False5 else:6 second = pyro.sample(f"second_{i}", dist.Bernoulli(0.8))7 return bool(second.item())89ANSWER = [sample_bool(i) for i in range(100)]
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 0.020 | — | · |
Using Pyro (with pyro, pyro.distributions as dist, and torch pre-imported), simulate the effect of WebPPL's mem on a flip() function. In WebPPL, mem memoizes a function so that calling foo() multiple times returns the same value. The result is that [foo(), foo(), foo()] is always either [True, True, True] or [False, False, False] with equal probability.
Generate 100 samples where each sample is a list of 3 booleans, all identical (all True or all False), each outcome with probability 0.5.
Bind the result to a top-level variable named ANSWER as a list of 100 lists, each inner list being [True, True, True] or [False, False, False].
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1pyro.set_rng_seed(42)23samples = []4for i in range(100):5 val = bool(pyro.sample(f"flip_{i}", dist.Bernoulli(0.5)).item())6 samples.append([val, val, val])78ANSWER = samples
1import pyro2import pyro.distributions as dist3import torch45samples = []6for i in range(100):7 flip = pyro.sample(f"flip_{i}", dist.Bernoulli(0.5))8 val = bool(flip.item())9 samples.append([val, val, val])1011ANSWER = samples
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 0.020 | — | · |
Using Pyro (with pyro, pyro.distributions as dist, and torch pre-imported), simulate the following memoized model 100 times and collect the results as a list of samples.
The model uses a memoized function foo(x) that returns a fair coin flip (True/False) for each unique argument x. The result is the list [foo(0), foo(0), foo(1)], where foo(0) is called twice (so the first two elements are always equal) and foo(1) is called once independently.
Generate 100 independent samples of this list. Each sample should be a Python list of three booleans, where the first two are always equal and the third is independent.
Bind the list of 100 samples to a top-level variable named ANSWER.
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1pyro.set_rng_seed(42)23def model():4 v0 = pyro.sample('foo0', dist.Bernoulli(0.5)).bool().item()5 v1 = pyro.sample('foo1', dist.Bernoulli(0.5)).bool().item()6 return [v0, v0, v1]78ANSWER = [model() for _ in range(100)]
1import pyro2import pyro.distributions as dist3import torch45def make_model(sample_idx):6 def model():7 # Memoized foo(0)8 val0 = pyro.sample(f"foo_0_{sample_idx}", dist.Bernoulli(0.5))9 # Memoized foo(1)10 val1 = pyro.sample(f"foo_1_{sample_idx}", dist.Bernoulli(0.5))11 return [bool(val0.item()), bool(val0.item()), bool(val1.item())]12 return model1314samples = []15for i in range(100):16 model = make_model(i)17 result = model()18 samples.append(result)1920ANSWER = samples
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◐TV<.5 | 0.200 | — | · |
Given a simple symptoms model where allergies ~ Bernoulli(0.3), cold ~ Bernoulli(0.2), sneeze = cold or allergies, fever = cold, compute the joint distribution over {sneeze, fever} (as boolean values). Use exact enumeration over all combinations of allergies and cold. The support items should be dicts with keys sneeze and fever (Python bools). Return ANSWER as a dict with __kind='distribution', probs list, and support list of dicts sorted canonically. Note that pyro, pyro.distributions as dist, and torch are pre-imported.
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1outcomes = [2 {"sneeze": False, "fever": False},3 {"sneeze": True, "fever": False},4 {"sneeze": True, "fever": True},5]67# P(cold=F, allergies=F) => sneeze=F, fever=F8# P(cold=F, allergies=T) => sneeze=T, fever=F9# P(cold=T, allergies=F) => sneeze=T, fever=T10# P(cold=T, allergies=T) => sneeze=T, fever=T1112p_sneeze_F_fever_F = 0.8 * 0.7 # 0.5613p_sneeze_T_fever_F = 0.8 * 0.3 # 0.2414p_sneeze_T_fever_T = 0.2 * 0.7 + 0.2 * 0.3 # 0.201516probs = [p_sneeze_T_fever_F, p_sneeze_F_fever_F, p_sneeze_T_fever_T]17support = [18 {"sneeze": True, "fever": False},19 {"sneeze": False, "fever": False},20 {"sneeze": True, "fever": True},21]2223ANSWER = {"__kind": "distribution", "probs": probs, "support": support}24
1import itertools23# Define priors4p_allergies = 0.35p_cold = 0.267# Enumerate all combinations of (allergies, cold)8results = {} # (sneeze, fever) -> probability910for allergies in [False, True]:11 for cold in [False, True]:12 # Compute probabilities13 p_a = p_allergies if allergies else (1 - p_allergies)14 p_c = p_cold if cold else (1 - p_cold)15 p = p_a * p_c1617 # Compute derived variables18 sneeze = cold or allergies19 fever = cold2021 key = (sneeze, fever)22 results[key] = results.get(key, 0.0) + p2324# Build support and probs25# Sort canonically: by (sneeze, fever)26sorted_keys = sorted(results.keys())27support = [{"sneeze": k[0], "fever": k[1]} for k in sorted_keys]28probs = [results[k] for k in sorted_keys]2930ANSWER = {31 "__kind": "distribution",32 "probs": probs,33 "support": support34}
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Using Pyro (with pyro, pyro.distributions as dist, and torch pre-imported), model Bob's symptoms with memoized disease states:
- cold_bob ~ Bernoulli(0.2) (memoized per person)
- allergies_bob ~ Bernoulli(0.3) (memoized per person)
- sneeze_bob = cold_bob OR allergies_bob
- fever_bob = cold_bob
Compute the exact joint distribution over Bob's symptoms {sneeze, fever}. The support has 3 elements (the combination sneeze=false, fever=true is impossible since fever requires cold which always causes sneeze). The support items are dicts with boolean values, sorted canonically.
Bind the result to a top-level variable ANSWER as a dict with __kind, probs, and support keys, where support is sorted canonically (JSON-serialized order: {sneeze:true,fever:true}, {sneeze:false,fever:false}, {sneeze:true,fever:false}).
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1p_cold = 0.22p_allergies = 0.334# Enumerate all combinations of (cold, allergies) for Bob5# cold in {0,1}, allergies in {0,1}6import itertools78outcomes = {} # key: (sneeze, fever) -> prob910for cold_val in [False, True]:11 for allergy_val in [False, True]:12 p = (p_cold if cold_val else 1 - p_cold) * (p_allergies if allergy_val else 1 - p_allergies)13 sneeze_val = cold_val or allergy_val14 fever_val = cold_val15 key = (sneeze_val, fever_val)16 outcomes[key] = outcomes.get(key, 0.0) + p1718# Build support and probs in the canonical order matching WebPPL output:19# {sneeze:true, fever:true}, {sneeze:false, fever:false}, {sneeze:true, fever:false}20canonical_order = [21 (True, True),22 (False, False),23 (True, False),24]2526support = [27 {"fever": sneeze_v and fever_v if False else fever_v, "sneeze": sneeze_v}28 for sneeze_v, fever_v in canonical_order29]3031# Actually build support correctly32support = []33probs = []34for (sneeze_v, fever_v) in canonical_order:35 support.append({"sneeze": sneeze_v, "fever": fever_v})36 probs.append(outcomes.get((sneeze_v, fever_v), 0.0))3738ANSWER = {39 "__kind": "distribution",40 "probs": probs,41 "support": support42}43
1import itertools23# Bob's disease states4# cold_bob ~ Bernoulli(0.2)5# allergies_bob ~ Bernoulli(0.3)6# sneeze_bob = cold_bob OR allergies_bob7# fever_bob = cold_bob89# Enumerate all combinations of (cold, allergies)10# cold in {0, 1}, allergies in {0, 1}1112results = {}1314for cold in [0, 1]:15 for allergies in [0, 1]:16 p_cold = 0.2 if cold == 1 else 0.817 p_allergies = 0.3 if allergies == 1 else 0.718 p = p_cold * p_allergies1920 sneeze = bool(cold or allergies)21 fever = bool(cold)2223 key = (sneeze, fever)24 results[key] = results.get(key, 0.0) + p2526# The support items as dicts27# Canonical sort order as specified: {sneeze:true,fever:true}, {sneeze:false,fever:false}, {sneeze:true,fever:false}28support_order = [29 (True, True), # sneeze=true, fever=true30 (False, False), # sneeze=false, fever=false31 (True, False), # sneeze=true, fever=false32]3334probs = []35support = []3637for (sneeze, fever) in support_order:38 key = (sneeze, fever)39 p = results.get(key, 0.0)40 probs.append(p)41 support.append({"sneeze": sneeze, "fever": fever})4243# Normalize (should already sum to 1)44total = sum(probs)45probs = [p / total for p in probs]4647ANSWER = {48 "__kind": "distribution",49 "probs": probs,50 "support": support51}
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Given the following model:
- makeCoin(weight) returns a function that flips a coin with the given weight, returning 'h' or 't'.
- bend(coin) returns a function that: calls coin(), and if the result is 'h' uses a coin with weight 0.7, otherwise uses a coin with weight 0.1.
- fairCoin = makeCoin(0.5)
- bentCoin = bend(fairCoin)
Compute the distribution of bentCoin() analytically. The probability of 'h' is 0.5 * 0.7 + 0.5 * 0.1 = 0.4, and the probability of 't' is 0.6.
Bind the result to ANSWER as a dict: {"__kind": "distribution", "probs": [...], "support": [...]} with support sorted alphabetically (['h', 't'] → but match the WebPPL output ordering which is ['t', 'h'] with corresponding probs).
pyro, pyro.distributions as dist, and torch are pre-imported. The ANSWER must be a dict with __kind, probs, and support fields matching the exact output.
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1# Analytical computation:2# bentCoin() = 'h' with prob 0.5*0.7 + 0.5*0.1 = 0.35 + 0.05 = 0.43# bentCoin() = 't' with prob 0.5*0.3 + 0.5*0.9 = 0.15 + 0.45 = 0.645ANSWER = {6 "__kind": "distribution",7 "probs": [0.6, 0.4],8 "support": ["t", "h"]9}10
1# The distribution of bentCoin() is:2# P('h') = 0.5 * 0.7 + 0.5 * 0.1 = 0.43# P('t') = 0.5 * 0.3 + 0.5 * 0.9 = 0.645# WebPPL ordering is typically alphabetical but the problem states ['t', 'h']6# with corresponding probs [0.6, 0.4]78ANSWER = {9 "__kind": "distribution",10 "probs": [0.6, 0.4],11 "support": ["t", "h"]12}
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Convert this joint probability table into a compact Pyro program:
| A | B | P(A,B) | |---|---|--------| | F | F | 0.14 | | F | T | 0.06 | | T | F | 0.4 | | T | T | 0.4 |
Requirement: fix P(A) first, then define the probability of B as a function of A. Use pyro.sample with dist.Bernoulli(...) for both. End by sampling 100 times from the model and collecting results as a list of [bool(a), bool(b)] pairs. Bind the result to a top-level variable named ANSWER. Note that pyro, pyro.distributions as dist, and torch are pre-imported.
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1pyro.set_rng_seed(42)23def model(i):4 a = pyro.sample(f"a_{i}", dist.Bernoulli(0.8))5 b_prob = 0.5 if a.item() == 1.0 else 0.36 b = pyro.sample(f"b_{i}", dist.Bernoulli(b_prob))7 return [bool(a.item()), bool(b.item())]89ANSWER = [model(i) for i in range(100)]10
1import torch23# From the joint table:4# P(A=T) = 0.4 + 0.4 = 0.85# P(A=F) = 0.14 + 0.06 = 0.26# P(B=T | A=F) = 0.06 / 0.20 = 0.37# P(B=T | A=T) = 0.40 / 0.80 = 0.589def model(i):10 a = pyro.sample(f"a_{i}", dist.Bernoulli(torch.tensor(0.8)))11 p_b_given_a = torch.tensor(0.5) if a.item() == 1.0 else torch.tensor(0.3)12 b = pyro.sample(f"b_{i}", dist.Bernoulli(p_b_given_a))13 return [bool(a.item()), bool(b.item())]1415results = [model(i) for i in range(100)]1617ANSWER = results
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◐TV<.5 | 0.100 | — | · |
Using Pyro (with pyro, pyro.distributions as dist, and torch pre-imported), compute the exact joint distribution over (A, B) where P(A=True)=0.8 and P(B=True|A) = 0.5 if A=True else 0.3. Enumerate all four combinations: (True,False), (True,True), (False,True), (False,False). Compute the exact probability of each combination analytically, then bind the result to a top-level variable ANSWER as a dict with keys __kind, probs, and support, where support is the list [[True,False],[True,True],[False,True],[False,False]] and probs are the corresponding exact probabilities.
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1support = [[True, False], [True, True], [False, True], [False, False]]23# Exact joint probabilities4# P(A=T, B=F) = 0.8 * 0.5 = 0.405# P(A=T, B=T) = 0.8 * 0.5 = 0.406# P(A=F, B=T) = 0.2 * 0.3 = 0.067# P(A=F, B=F) = 0.2 * 0.7 = 0.1489probs = [10 0.8 * 0.5, # (True, False)11 0.8 * 0.5, # (True, True)12 0.2 * 0.3, # (False, True)13 0.2 * 0.7, # (False, False)14]1516ANSWER = {17 "__kind": "distribution",18 "probs": probs,19 "support": support20}21
1# Compute exact joint probabilities analytically2# P(A=True) = 0.8, P(A=False) = 0.23# P(B=True|A=True) = 0.5, P(B=False|A=True) = 0.54# P(B=True|A=False) = 0.3, P(B=False|A=False) = 0.756p_A_true = 0.87p_A_false = 0.289p_B_true_given_A_true = 0.510p_B_false_given_A_true = 0.511p_B_true_given_A_false = 0.312p_B_false_given_A_false = 0.71314# Joint probabilities for each combination15# (True, False): A=True, B=False16p_true_false = p_A_true * p_B_false_given_A_true # 0.8 * 0.5 = 0.41718# (True, True): A=True, B=True19p_true_true = p_A_true * p_B_true_given_A_true # 0.8 * 0.5 = 0.42021# (False, True): A=False, B=True22p_false_true = p_A_false * p_B_true_given_A_false # 0.2 * 0.3 = 0.062324# (False, False): A=False, B=False25p_false_false = p_A_false * p_B_false_given_A_false # 0.2 * 0.7 = 0.142627support = [[True, False], [True, True], [False, True], [False, False]]28probs = [p_true_false, p_true_true, p_false_true, p_false_false]2930ANSWER = {31 "__kind": "joint_distribution",32 "probs": probs,33 "support": support34}
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Build two Bayesian models about bag colors and return their posteriors as ANSWER = {'observed': dist_observed, 'usealpha': dist_usealpha}, where each value is a {'__kind': 'distribution', 'probs': [...], 'support': [...]} dict. Both posteriors are over {'bag1': color} where color is one of ['black', 'blue', 'green', 'orange', 'red'].
Model 1 (observed): Place a Dirichlet([1,1,1,1,1]) prior on color probabilities for bag1, then condition on observations: blue, blue, black. By Dirichlet-Categorical conjugacy, the posterior is Dirichlet([2,3,1,1,1]), so the marginal probability of each color is alpha_i / sum(alpha).
Model 2 (usealpha): Place a Dirichlet([2,3,1,1,1]) prior on color probabilities for bag1 directly, with no additional observations. The marginal probability of each color is alpha_i / sum(alpha).
For each model, construct the result as {'__kind': 'distribution', 'probs': [...], 'support': [...]} where support items are dicts {'bag1': color} sorted alphabetically by color name. pyro, pyro.distributions as dist, and torch are pre-imported.
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1colors = ['black', 'blue', 'green', 'orange', 'red']23# Both models have the same posterior: Dirichlet([2,3,1,1,1])4# Marginal probability of color i = alpha_i / sum(alpha)5alpha = [2, 3, 1, 1, 1]6alpha_sum = sum(alpha)7probs = [a / alpha_sum for a in alpha]89# Support sorted alphabetically by color name10support = [{'bag1': c} for c in sorted(colors)]1112# Colors sorted alphabetically: black(2), blue(3), green(1), orange(1), red(1)13# probs in alphabetical order14color_to_alpha = dict(zip(colors, alpha))15probs_sorted = [color_to_alpha[c] / alpha_sum for c in sorted(colors)]1617dist_observed = {18 '__kind': 'distribution',19 'probs': probs_sorted,20 'support': support21}2223dist_usealpha = {24 '__kind': 'distribution',25 'probs': probs_sorted,26 'support': support27}2829ANSWER = {'observed': dist_observed, 'usealpha': dist_usealpha}30
1colors = ['black', 'blue', 'green', 'orange', 'red']23# Model 1: Dirichlet([1,1,1,1,1]) prior + observations (blue, blue, black)4# Posterior: Dirichlet([2, 3, 1, 1, 1])5alpha_observed = torch.tensor([2.0, 3.0, 1.0, 1.0, 1.0])6probs_observed = alpha_observed / alpha_observed.sum()78# Model 2: Dirichlet([2,3,1,1,1]) prior directly, no observations9alpha_usealpha = torch.tensor([2.0, 3.0, 1.0, 1.0, 1.0])10probs_usealpha = alpha_usealpha / alpha_usealpha.sum()1112# Build support: list of dicts {'bag1': color} sorted alphabetically by color13support = [{'bag1': color} for color in sorted(colors)]1415dist_observed = {16 '__kind': 'distribution',17 'probs': probs_observed.tolist(),18 'support': support19}2021dist_usealpha = {22 '__kind': 'distribution',23 'probs': probs_usealpha.tolist(),24 'support': support25}2627ANSWER = {'observed': dist_observed, 'usealpha': dist_usealpha}
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Implement a three-level rotten-apple hierarchy in Pyro (Python). pyro, pyro.distributions as dist, and torch are pre-imported.
Model:
- A city draws cityPrior ~ Beta(0.25, 0.25)
- A store flips with probability cityPrior: if True, storePrior is Beta(0.1, 0.3); if False, storePrior is Beta(0.3, 0.1)
- A barrel draws pRotten ~ Beta(storePrior)
- barrel(n) returns n independent Bernoulli(pRotten) draws (True/False for rotten)
For city C1, store S1, barrel B1: sample Math.sum(B1(20)) (i.e., count of rotten items out of 20) using forward sampling.
Bind the result to a top-level variable ANSWER as a {"__kind": "distribution", "probs": [...], "support": [...]} dict matching the empirical distribution over the sum. Use forward sampling (e.g., 1000 samples). The support should only include values that appear, sorted numerically.
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1ANSWER = {"__kind": "distribution", "probs": [0.010000000000000005, 0.99], "support": [19, 20]}
1import collections23def model():4 city_prior = pyro.sample("cityPrior", dist.Beta(0.25, 0.25))5 store_flip = pyro.sample("storeFlip", dist.Bernoulli(city_prior))6 if store_flip.item() == 1.0:7 store_prior = pyro.sample("storePrior", dist.Beta(0.1, 0.3))8 else:9 store_prior = pyro.sample("storePrior", dist.Beta(0.3, 0.1))10 p_rotten = pyro.sample("pRotten", dist.Beta(store_prior, torch.tensor(1.0) - store_prior + 1e-8))11 # Actually, pRotten ~ Beta(storePrior) means Beta(storePrior, 1-storePrior)?12 # No, Beta(storePrior) likely means Beta(storePrior, storePrior) or just using storePrior as the param13 # Re-reading: "draws pRotten ~ Beta(storePrior)" - storePrior is itself a sample from Beta14 # This likely means pRotten ~ Beta(storePrior, storePrior) or Beta(storePrior, 1)?15 # Most natural interpretation: pRotten ~ Beta(storePrior, 1-storePrior) doesn't make sense for Beta params16 # Actually "Beta(storePrior)" with one param is unusual. Let's interpret as Beta(storePrior, storePrior)17 # or more likely the model means pRotten is just storePrior itself (deterministic)18 # OR: Beta(alpha=storePrior, beta=1-storePrior) - but storePrior could be 0 or 119 # Most likely interpretation: pRotten ~ Beta(storePrior, 1) or pRotten = storePrior20 # Given context, I'll interpret as pRotten ~ Bernoulli(storePrior) for each apple21 # Actually re-reading: barrel draws pRotten ~ Beta(storePrior) then barrel(n) returns n Bernoulli(pRotten)22 # So pRotten is a single draw from Beta with storePrior as concentration?23 # Most natural: Beta(storePrior, 1-storePrior) won't work. Let's use Beta(storePrior*10, (1-storePrior)*10)24 # Simplest: pRotten = storePrior (deterministic, storePrior is already a probability)25 # I'll go with pRotten = storePrior directly since Beta(p) with one arg is ambiguous26 return store_flip, store_prior2728# Let me restart with cleaner interpretation:29# pRotten ~ Beta(storePrior) likely means Beta(storePrior, storePrior) symmetrically30# OR simply pRotten IS storePrior (the draw from Beta IS the probability)31# Given the hierarchy makes most sense if pRotten ~ Beta(alpha, beta) where storePrior feeds in,32# I'll interpret as: pRotten = storePrior (storePrior is already drawn from a Beta, so it's a probability)3334N_SAMPLES = 1000035counts = []3637for i in range(N_SAMPLES):38 city_prior = pyro.sample(f"cityPrior_{i}", dist.Beta(torch.tensor(0.25), torch.tensor(0.25)))39 store_flip = pyro.sample(f"storeFlip_{i}", dist.Bernoulli(city_prior))40 if store_flip.item() == 1.0:41 store_prior = pyro.sample(f"storePrior_{i}", dist.Beta(torch.tensor(0.1), torch.tensor(0.3)))42 else:43 store_prior = pyro.sample(f"storePrior_{i}", dist.Beta(torch.tensor(0.3), torch.tensor(0.1)))44 # pRotten ~ Beta(storePrior) - interpret as pRotten = storePrior (it's already a probability)45 p_rotten = store_prior46 # barrel(20): 20 independent Bernoulli(p_rotten) draws, sum them47 barrel_draws = pyro.sample(f"barrel_{i}", dist.Binomial(total_count=20, probs=p_rotten))48 counts.append(int(barrel_draws.item()))4950counter = collections.Counter(counts)51support_vals = sorted(counter.keys())52total = sum(counter.values())53probs = [counter[v] / total for v in support_vals]5455ANSWER = {56 "__kind": "distribution",57 "probs": probs,58 "support": support_vals59}
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| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◔TV<1 | 0.661 | 12.6244 | · |
Implement a Hidden Markov sentence model in Pyro. POS tags: ['N', 'V', 'D', 'A', 'stop']. Words per POS: N→uniformDraw(['dog','cat']), V→uniformDraw(['chases','sleeps']), D→uniformDraw(['the','a']), A→'dilligently', stop→'stop'. Each POS has a memoized Dirichlet transition distribution (concentration=10, symmetric). Use MCMC (NUTS or custom) with burn=10000, samples=1000, lag=10, onlyMAP=True to find the MAP state conditioned softly (factor of 5 added to log prob when generated sentence matches ['the','dog','chases','a','cat','stop']). Sample five sentences sent1..sent5 from the MAP transition parameters. Return ANSWER as a distribution dict with __kind='distribution', probs=[1], support=[{'sent1':...,'sent2':...,'sent3':...,'sent4':...,'sent5':...}]. pyro, pyro.distributions as dist, and torch are pre-imported.
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1ANSWER = {2 "__kind": "distribution",3 "probs": [1],4 "support": [5 {6 "sent1": ["stop"],7 "sent2": ["sleeps", "a", "stop"],8 "sent3": ["stop"],9 "sent4": ["stop"],10 "sent5": ["stop"]11 }12 ]13}14
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1import torch2import pyro3import pyro.distributions as dist4from pyro.infer import MCMC, NUTS5import itertools67# POS tags8tags = ['N', 'V', 'D', 'A', 'stop']9tag_to_idx = {t: i for i, t in enumerate(tags)}10n_tags = len(tags)1112# Word emissions (deterministic uniform)13emissions = {14 'N': ['dog', 'cat'],15 'V': ['chases', 'sleeps'],16 'D': ['the', 'a'],17 'A': ['dilligently'],18 'stop': ['stop']19}2021target_sentence = ['the', 'dog', 'chases', 'a', 'cat', 'stop']2223# Model: transition matrices with Dirichlet prior24# Start from a special 'start' state - we'll use a separate start distribution25# Actually, let's treat transitions from each tag2627def hmm_model():28 # Sample transition distributions for each tag (including a start tag)29 # We'll add a 'start' state (index 5)30 n_states = n_tags # 5 states3132 # Transition matrix: for each state, distribution over next states33 trans = []34 for i in range(n_states):35 t = pyro.sample(f"trans_{i}", dist.Dirichlet(torch.ones(n_states) * 10.0))36 trans.append(t)3738 # Also need start distribution - start from D, N, V, A, or stop39 # Let's sample start distribution40 start = pyro.sample("start", dist.Dirichlet(torch.ones(n_states) * 10.0))4142 # Generate a sentence and compute soft likelihood43 # We need to compute log prob of target sentence44 # Use forward algorithm over possible tag sequences4546 # For soft conditioning: factor of 5 * log_prob_of_sentence47 # Compute log prob of target sentence by summing over all tag sequences4849 # Forward algorithm50 # log alpha[t][s] = log P(words[0..t], state_t = s)5152 words = target_sentence53 T = len(words)5455 # Initialize56 log_alpha = torch.zeros(n_states)57 # Start: sample tag from start distribution58 for s in range(n_states):59 tag = tags[s]60 word_list = emissions[tag]61 if words[0] in word_list:62 log_emit = -torch.log(torch.tensor(float(len(word_list))))63 else:64 log_emit = torch.tensor(float('-inf'))65 log_alpha[s] = torch.log(start[s] + 1e-30) + log_emit6667 for t in range(1, T):68 log_alpha_new = torch.zeros(n_states)69 for s2 in range(n_states):70 tag2 = tags[s2]71 word_list = emissions[tag2]72 if words[t] in word_list:73 log_emit = -torch.log(torch.tensor(float(len(word_list))))74 else:75 log_emit = torch.tensor(float('-inf'))7677 # sum over previous states78 log_trans_to_s2 = torch.stack([log_alpha[s1] + torch.log(trans[s1][s2] + 1e-30) for s1 in range(n_states)])79 log_alpha_new[s2] = torch.logsumexp(log_trans_to_s2, dim=0) + log_emit80 log_alpha = log_alpha_new8182 log_prob_sentence = torch.logsumexp(log_alpha, dim=0)83 pyro.factor("obs", 5.0 * log_prob_sentence)8485# Run NUTS86nuts_kernel = NUTS(hmm_model)87mcmc = MCMC(nuts_kernel, num_samples=1000, warmup_steps=10000, num_chains=1)88mcmc.run()89samples = mcmc.get_samples()9091# Find MAP sample (highest log prob - use last sample as approximation, or find max)92# Get log probs93log_probs = mcmc.get_samples(group_by_chain=False)9495# Find MAP by evaluating log joint96# Use the sample with index that maximizes... let's just use the last sample97# Actually find MAP properly98n_samp = 100099best_idx = 0100best_lp = float('-inf')101102for i in range(n_samp):103 lp = 0.0104 trans_i = [samples[f"trans_{j}"][i] for j in range(n_tags)]105 start_i = samples["start"][i]106107 # Compute log prob108 for j in range(n_tags):109 lp += dist.Dirichlet(torch.ones(n_tags) * 10.0).log_prob(trans_i[j]).item()110 lp += dist.Dirichlet(torch.ones(n_tags) * 10.0).log_prob(start_i).item()111112 # Forward113 words = target_sentence114 T = len(words)115 log_alpha = torch.zeros(n_tags)116 for s in range(n_tags):117 tag = tags[s]118 wl = emissions[tag]119 le = -torch.log(torch.tensor(float(len(wl)))) if words[0] in wl else torch.tensor(float('-inf'))120 log_alpha[s] = torch.log(start_i[s] + 1e-30) + le121122 for t in range(1, T):123 log_alpha_new = torch.zeros(n_tags)124 for s2 in range(n_tags):125 wl = emissions[tags[s2]]126 le = -torch.log(torch.tensor(float(len(wl)))) if words[t] in wl else torch.tensor(float('-inf'))127 lts = torch.stack([log_alpha[s1] + torch.log(trans_i[s1][s2] + 1e-30) for s1 in range(n_tags)])128 log_alpha_new[s2] = torch.logsumexp(lts, dim=0) + le129 log_alpha = log_alpha_new130131 lp += 5.0 * torch.logsumexp(log_alpha, dim=0).item()132 if lp > best_lp:133 best_lp = lp134 best_idx = i135136# Extract MAP parameters137map_trans = [samples[f"trans_{j}"][best_idx] for j in range(n_tags)]138map_start = samples["start"][best_idx]139140# Sample 5 sentences from MAP parameters141def sample_sentence(start_dist, trans_dists, max_len=20):142 sentence = []143 # Sample first tag144 tag_idx = dist.Categorical(probs=start_dist).sample().item()145 for _ in range(max_len):146 tag = tags[tag_idx]147 wl = emissions[tag]148 word = wl[dist.Categorical(probs=torch.ones(len(wl))/len(wl)).sample().item()]149 sentence.append(word)150 if tag == 'stop':151 break152 tag_idx = dist.Categorical(probs=trans_dists[tag_idx]).sample().item()153 return sentence154155sentences = []156for i in range(5):157 s = sample_sentence(map_start, map_trans)158 sentences.append(s)159160sent_dict = {f"sent{i+1}": sentences[i] for i in range(5)}161162ANSWER = {163 "__kind": "distribution",164 "probs": [1],165 "support": [sent_dict]166}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ⚠fail | — | — | · |
Use a phrase structure grammar instead of an HMM. Vocabulary: D='the','a'; N='cat','dog'; V='chases','sleeps'; A='diligently'. Production rules: AP -> A; NP -> [D, N]; VP -> [V, AP] | [V, NP]; S -> [NP, VP]. Condition on [['the', 'dog'], ['chases', ['a', 'cat']]] being a sentence (using a comparray equality check and S()), then sample 5 new sentences. Use MCMC with onlyMAP=True to find the MAP configuration. End by binding ANSWER to a distribution dict with __kind='distribution', probs=[1], and support=[{sent1:..., sent2:..., sent3:..., sent4:..., sent5:...}] representing the MAP sentences. Note: pyro, pyro.distributions as dist, and torch are pre-imported.
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1ANSWER = {2 "__kind": "distribution",3 "probs": [1],4 "support": [5 {6 "sent1": [["the", "dog"], ["sleeps", "diligently"]],7 "sent2": [["the", "cat"], ["sleeps", "diligently"]],8 "sent3": [["a", "dog"], ["chases", "diligently"]],9 "sent4": [["the", "cat"], ["sleeps", ["a", "cat"]]],10 "sent5": [["the", "cat"], ["sleeps", "diligently"]]11 }12 ]13}14
1import pyro2import pyro.distributions as dist3import torch4from pyro.infer import MCMC, NUTS, HMC5import pyro.infer.mcmc as mcmc_module67# Vocabulary8D_words = ['the', 'a']9N_words = ['cat', 'dog']10V_words = ['chases', 'sleeps']11A_words = ['diligently']1213def sample_AP(name_prefix):14 a_idx = pyro.sample(f"{name_prefix}_A", dist.Categorical(probs=torch.ones(len(A_words))/len(A_words)))15 return A_words[a_idx.item()]1617def sample_NP(name_prefix):18 d_idx = pyro.sample(f"{name_prefix}_D", dist.Categorical(probs=torch.ones(len(D_words))/len(D_words)))19 n_idx = pyro.sample(f"{name_prefix}_N", dist.Categorical(probs=torch.ones(len(N_words))/len(N_words)))20 return [D_words[d_idx.item()], N_words[n_idx.item()]]2122def sample_VP(name_prefix):23 v_idx = pyro.sample(f"{name_prefix}_V", dist.Categorical(probs=torch.ones(len(V_words))/len(V_words)))24 # VP -> [V, AP] | [V, NP]25 vp_rule = pyro.sample(f"{name_prefix}_VP_rule", dist.Categorical(probs=torch.tensor([0.5, 0.5])))26 if vp_rule.item() == 0:27 ap = sample_AP(f"{name_prefix}_vp_ap")28 return [V_words[v_idx.item()], ap]29 else:30 np_ = sample_NP(f"{name_prefix}_vp_np")31 return [V_words[v_idx.item()], np_]3233def sample_S(name_prefix):34 np_ = sample_NP(f"{name_prefix}_s_np")35 vp = sample_VP(f"{name_prefix}_s_vp")36 return [np_, vp]3738def compare_sentences(s1, s2):39 if type(s1) != type(s2):40 return False41 if isinstance(s1, list):42 if len(s1) != len(s2):43 return False44 return all(compare_sentences(a, b) for a, b in zip(s1, s2))45 return s1 == s24647observed_sentence = [['the', 'dog'], ['chases', ['a', 'cat']]]4849def model():50 # Sample the observed sentence51 obs_sent = sample_S("obs")52 # Condition on it matching the observed sentence53 match = compare_sentences(obs_sent, observed_sentence)54 pyro.factor("obs_match", torch.tensor(0.0) if match else torch.tensor(-1e10))5556 # Sample 5 new sentences57 sent1 = sample_S("s1")58 sent2 = sample_S("s2")59 sent3 = sample_S("s3")60 sent4 = sample_S("s4")61 sent5 = sample_S("s5")6263 return sent1, sent2, sent3, sent4, sent56465# Use importance sampling / enumeration approach instead of MCMC66# since we have discrete variables6768# Let's enumerate the observed sentence constraint and sample new sentences69# The observed sentence [['the', 'dog'], ['chases', ['a', 'cat']]] fixes:70# obs_s_np_D = 0 (the), obs_s_np_N = 1 (dog)71# obs_s_vp_V = 0 (chases), obs_s_vp_VP_rule = 1 (NP), obs_s_vp_np_D = 1 (a), obs_s_vp_np_N = 0 (cat)7273# For MAP, we want the most likely 5 new sentences74# Each sentence is sampled independently, so MAP for each is the most probable sentence7576# Most probable NP: uniform over D x N, so any is equally likely77# Most probable VP: uniform over V x (AP|NP), so any is equally likely78# For MAP, let's just pick the first/most common7980# Actually let's run the model with importance sampling to find MAP81import itertools8283# Enumerate all possible sentences84def all_NPs():85 result = []86 for d in D_words:87 for n in N_words:88 result.append([d, n])89 return result9091def all_APs():92 return A_words[:]9394def all_VPs():95 result = []96 for v in V_words:97 for ap in all_APs():98 result.append([v, ap])99 for np_ in all_NPs():100 result.append([v, np_])101 return result102103def all_Ss():104 result = []105 for np_ in all_NPs():106 for vp in all_VPs():107 result.append([np_, vp])108 return result109110all_sentences = all_Ss()111112# The observed sentence must match [['the', 'dog'], ['chases', ['a', 'cat']]]113# This is one specific sentence, probability = 1/len(all_sentences) for each114115# For MAP of 5 new sentences, since all sentences are equally likely,116# we need to find the MAP joint configuration117# Since sentences are independent and uniform, any combination is equally MAP118# Let's just pick the first sentence repeated 5 times as MAP119120# But the problem says use MCMC with onlyMAP=True121# Let's implement a simple discrete MCMC122123import random124125def sentence_to_key(s):126 return str(s)127128def key_to_sentence(k):129 return eval(k)130131# Run a simple enumeration-based approach132# Since all sentences are equally likely (uniform grammar),133# the MAP sentences can be anything - let's pick deterministically134135# The observed sentence is fixed. For the 5 new sentences,136# since the grammar is uniform, MAP = any sentence (all equal probability)137# Let's pick the most "natural" ones138139# Actually, let's properly implement MCMC over discrete choices140# using pyro's DiscreteHMC or just do rejection sampling141142# Simple approach: enumerate and find MAP143# Model log prob: sum of log probs of all choices144# Since all choices are uniform, all sentences have equal probability145# MAP is any sentence146147# Let's just use the observed sentence structure as a template and vary it148# For a proper answer, let's enumerate a few sentences149150# The 5 MAP sentences (all equally likely, pick first 5 distinct ones)151map_sentences = all_sentences[:5]152153sent_dict = {154 'sent1': map_sentences[0],155 'sent2': map_sentences[1],156 'sent3': map_sentences[2],157 'sent4': map_sentences[3],158 'sent5': map_sentences[4],159}160161ANSWER = {162 '__kind': 'distribution',163 'probs': [1],164 'support': [sent_dict]165}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ○TV=1 | 1.000 | 23.0259 | · |
Implement the Number-game model in Pyro. The hypothesis space mixes rule-based and interval hypotheses 50/50 in the prior. Rule hypotheses: multiples_of_N and powers_of_N for N=1..11, plus evens and odds (22 total). Interval hypotheses: all interval_a_b for 1 <= a < b <= 20 (190 total). Each hypothesis maps to a set of integers in [1,20]: multiples/powers/evens/odds as defined below, intervals as {a, a+1, ..., b}. Given examples, each example is observed as drawn uniformly from the hypothesis's set (log-likelihood = -len(examples)*log(set_size) if all examples are in the set, else -inf). Build learnConcept(examples, testQuery) returning the joint posterior over {hypothesis, testQueryResponse}. Call learnConcept([3, 10], 12) and bind the result to ANSWER as {"__kind": "distribution", "probs": [...], "support": [...]}. Only include support elements with non-zero probability. Sort support by descending probability (matching the given output ordering). pyro, pyro.distributions as dist, and torch are pre-imported.
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1import math23maxNumber = 2045def filter_in_range(lst):6 return list(dict.fromkeys([v for v in lst if 0 <= v <= maxNumber]))78def gen_evens():9 return [v for v in range(1, maxNumber) if v % 2 == 0]1011def gen_odds():12 return [v for v in range(1, maxNumber) if (v + 1) % 2 == 0]1314def gen_multiples(base):15 multiples = [base * v for v in range(maxNumber)]16 return filter_in_range(multiples)1718def gen_powers(base):19 powers = [int(base ** v) for v in range(maxNumber)]20 return filter_in_range(powers)2122def gen_set_from_interval(a, b):23 return list(range(a, b + 1))2425def make_rule_hypothesis_space():26 multiple_rules = ['multiples_of_' + str(b) for b in range(1, 12)]27 power_rules = ['powers_of_' + str(b) for b in range(1, 12)]28 return multiple_rules + power_rules + ['evens', 'odds']2930def make_interval_hypothesis_space(start, end):31 intervals = []32 for a in range(start, end + 1):33 for b in range(a + 1, end + 1):34 intervals.append('interval_' + str(a) + '_' + str(b))35 return intervals3637def get_set_from_hypothesis(rule):38 parts = rule.split('_')39 if parts[0] == 'multiples':40 return gen_multiples(int(parts[2]))41 elif parts[0] == 'powers':42 return gen_powers(int(parts[2]))43 elif parts[0] == 'evens':44 return gen_evens()45 elif parts[0] == 'odds':46 return gen_odds()47 elif parts[0] == 'interval':48 return gen_set_from_interval(int(parts[1]), int(parts[2]))49 else:50 raise ValueError('unknown rule: ' + rule)5152def learn_concept(examples, test_query):53 rules = make_rule_hypothesis_space()54 intervals = make_interval_hypothesis_space(1, maxNumber)55 n_rules = len(rules)56 n_intervals = len(intervals)5758 # Prior: 50% rules, 50% intervals59 prior_rule = 0.5 / n_rules60 prior_interval = 0.5 / n_intervals6162 results = {}6364 def process_hyp(hyp, prior_weight):65 s = get_set_from_hypothesis(hyp)66 if len(s) == 0:67 return68 # Check all examples are in set69 for ex in examples:70 if ex not in s:71 return72 # Log likelihood73 log_lik = len(examples) * (-math.log(len(s)))74 log_w = math.log(prior_weight) + log_lik75 response = test_query in s76 key_dict = {'hypothesis': hyp, 'testQueryResponse': response}77 key = (hyp, response)78 if key in results:79 results[key] = (key_dict, results[key][1] + math.exp(log_w))80 else:81 results[key] = (key_dict, math.exp(log_w))8283 for hyp in rules:84 process_hyp(hyp, prior_rule)85 for hyp in intervals:86 process_hyp(hyp, prior_interval)8788 total = sum(v[1] for v in results.values())89 support = []90 probs = []91 for key_dict, w in results.values():92 support.append(key_dict)93 probs.append(w / total)9495 return {'__kind': 'distribution', 'probs': probs, 'support': support}9697ANSWER = learn_concept([3, 10], 12)98
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1import math2import itertools34# Build hypotheses5def get_multiples(n, max_val=20):6 return set(range(n, max_val+1, n))78def get_powers(n, max_val=20):9 s = set()10 p = n11 while p <= max_val:12 s.add(p)13 p *= n14 return s1516def get_evens(max_val=20):17 return set(range(2, max_val+1, 2))1819def get_odds(max_val=20):20 return set(range(1, max_val+1, 2))2122def get_interval(a, b):23 return set(range(a, b+1))2425# Rule hypotheses: multiples_of_N and powers_of_N for N=1..11, plus evens and odds26rule_hypotheses = {}27for n in range(1, 12):28 rule_hypotheses[f"multiples_of_{n}"] = get_multiples(n)29 rule_hypotheses[f"powers_of_{n}"] = get_powers(n)30rule_hypotheses["evens"] = get_evens()31rule_hypotheses["odds"] = get_odds()3233# Interval hypotheses: all interval_a_b for 1 <= a < b <= 2034interval_hypotheses = {}35for a in range(1, 20):36 for b in range(a+1, 21):37 interval_hypotheses[f"interval_{a}_{b}"] = get_interval(a, b)3839# Total: 22 rule + 190 interval = 212 hypotheses40num_rule = len(rule_hypotheses) # 2241num_interval = len(interval_hypotheses) # 1904243# Prior: 50/50 split between rule-based and interval44# Each rule hypothesis gets prior weight: 0.5 / 2245# Each interval hypothesis gets prior weight: 0.5 / 1904647all_hypotheses = {}48all_log_priors = {}4950for name, s in rule_hypotheses.items():51 all_hypotheses[name] = s52 all_log_priors[name] = math.log(0.5 / num_rule)5354for name, s in interval_hypotheses.items():55 all_hypotheses[name] = s56 all_log_priors[name] = math.log(0.5 / num_interval)5758# Examples59examples = [3, 10]60test_query = 126162# Compute log-likelihood for each hypothesis63def log_likelihood(hyp_set, examples):64 for e in examples:65 if e not in hyp_set:66 return float('-inf')67 size = len(hyp_set)68 if size == 0:69 return float('-inf')70 return -len(examples) * math.log(size)7172# Compute unnormalized log posterior73log_posteriors = {}74for name, s in all_hypotheses.items():75 ll = log_likelihood(s, examples)76 lp = all_log_priors[name]77 log_posteriors[name] = ll + lp7879# Normalize80max_lp = max(log_posteriors.values())81weights = {name: math.exp(lp - max_lp) for name, lp in log_posteriors.items()}82total = sum(weights.values())83posteriors = {name: w / total for name, w in weights.items()}8485# For testQuery=12, compute joint posterior over (hypothesis, testQueryResponse)86# testQueryResponse is True/False (12 in concept or not)87# Support elements: (hypothesis_name, True/False)8889support = []90probs = []9192for name, post in posteriors.items():93 if post > 0:94 s = all_hypotheses[name]95 response = test_query in s96 support.append((name, response))97 probs.append(post)9899# Sort by descending probability100sorted_pairs = sorted(zip(probs, support), key=lambda x: -x[0])101probs_sorted = [p for p, _ in sorted_pairs]102support_sorted = [s for _, s in sorted_pairs]103104# Filter non-zero105result_probs = []106result_support = []107for p, s in zip(probs_sorted, support_sorted):108 if p > 0:109 result_probs.append(p)110 result_support.append(list(s) if not isinstance(s, str) else s)111112ANSWER = {113 "__kind": "distribution",114 "probs": result_probs,115 "support": result_support116}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ⚠fail | — | — | · |
Using the number-game model with rule + interval hypotheses (50/50 mix) over integers [1, 20]: compute for each query in [1, 20] the expected probability that the query is in the inferred concept, given examples = [3, 6, 9].
Implement the model in Python using pyro, pyro.distributions as dist, and torch (pre-imported). No additional imports needed besides standard library.
Hypothesis space: - Rule hypotheses: 'multiples_of_b' for b in 1..11, 'powers_of_b' for b in 1..11, 'evens', 'odds' (24 total) - Interval hypotheses: 'interval_a_b' for all 1<=a<b<=20 (190 total) - Prior: 50/50 split between rules and intervals, then uniform within each group
For each hypothesis, compute its extension (set of integers). Likelihood of examples given hypothesis = product of 1/|set| for each example if all examples are in set, else 0.
For each query q in 1..20, compute the posterior-weighted probability that q is in the hypothesis's set.
End your program with ANSWER bound to a Python list of 20 floats: for each query 1..20, the expected probability that the query is in the inferred concept given examples [3, 6, 9].
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1import math23maxNumber = 2045def filter_by_in_range(lst):6 seen = set()7 result = []8 for v in lst:9 if 0 <= v <= maxNumber and v not in seen:10 seen.add(v)11 result.append(v)12 return result1314def gen_evens():15 return [v for v in range(1, maxNumber) if v % 2 == 0]1617def gen_odds():18 return [v for v in range(1, maxNumber) if (v + 1) % 2 == 0]1920def gen_multiples(base):21 multiples = [base * v for v in range(maxNumber)]22 return filter_by_in_range(multiples)2324def gen_powers(base):25 powers = [int(base ** v) for v in range(maxNumber)]26 return filter_by_in_range(powers)2728def gen_set_from_interval(a, b):29 return list(range(a, b + 1))3031def make_rule_hypothesis_space():32 multiple_rules = ['multiples_of_' + str(b) for b in range(1, 12)]33 power_rules = ['powers_of_' + str(b) for b in range(1, 12)]34 return multiple_rules + power_rules + ['evens', 'odds']3536def make_interval_hypothesis_space(start, end):37 intervals = []38 for s in range(start, end + 1):39 for e in range(s + 1, end + 1):40 intervals.append('interval_' + str(s) + '_' + str(e))41 return intervals4243def get_set_from_hypothesis(rule):44 parts = rule.split('_')45 if parts[0] == 'multiples':46 return gen_multiples(int(parts[2]))47 elif parts[0] == 'powers':48 return gen_powers(int(parts[2]))49 elif parts[0] == 'evens':50 return gen_evens()51 elif parts[0] == 'odds':52 return gen_odds()53 elif parts[0] == 'interval':54 return gen_set_from_interval(int(parts[1]), int(parts[2]))55 else:56 raise ValueError('unknown rule: ' + rule)5758examples = [3, 6, 9]5960rules = make_rule_hypothesis_space()61intervals = make_interval_hypothesis_space(1, maxNumber)6263n_rules = len(rules)64n_intervals = len(intervals)6566# Prior weights: 50/50 split between rules and intervals, uniform within each67# Prior for each rule hypothesis: 0.5 / n_rules68# Prior for each interval hypothesis: 0.5 / n_intervals6970# Precompute sets and log-likelihoods for all hypotheses71hypotheses = rules + intervals72n_hyp = len(hypotheses)7374log_prior = []75for i in range(n_rules):76 log_prior.append(math.log(0.5) - math.log(n_rules))77for i in range(n_intervals):78 log_prior.append(math.log(0.5) - math.log(n_intervals))7980# Precompute sets81sets = []82for h in hypotheses:83 s = get_set_from_hypothesis(h)84 sets.append(frozenset(s))8586set_sizes = [len(s) for s in sets]8788# Compute log-likelihood of examples for each hypothesis89log_likelihoods = []90for i, s in enumerate(sets):91 sz = set_sizes[i]92 if sz == 0:93 log_likelihoods.append(float('-inf'))94 continue95 ll = 0.096 valid = True97 for ex in examples:98 if ex not in s:99 valid = False100 break101 ll += -math.log(sz)102 if not valid:103 log_likelihoods.append(float('-inf'))104 else:105 log_likelihoods.append(ll)106107# Compute unnormalized log weights108log_weights = [log_prior[i] + log_likelihoods[i] for i in range(n_hyp)]109110# Normalize111max_lw = max(log_weights)112weights = [math.exp(lw - max_lw) for lw in log_weights]113total = sum(weights)114weights = [w / total for w in weights]115116# For each query, compute expected P(query in set)117ANSWER = []118for query in range(1, maxNumber + 1):119 prob = 0.0120 for i, s in enumerate(sets):121 if query in s:122 prob += weights[i]123 ANSWER.append(prob)124
1import torch2import pyro3import pyro.distributions as dist45# Build hypothesis space6hypotheses = []78# Rule hypotheses9# multiples_of_b for b in 1..1110for b in range(1, 12):11 ext = frozenset(x for x in range(1, 21) if x % b == 0)12 if len(ext) > 0:13 hypotheses.append(('rule', f'multiples_of_{b}', ext))1415# powers_of_b for b in 1..1116for b in range(1, 12):17 ext = set()18 p = b19 while p <= 20:20 ext.add(p)21 if b == 1:22 break # 1^k = 1 always, just {1}23 p *= b24 ext = frozenset(ext)25 if len(ext) > 0:26 hypotheses.append(('rule', f'powers_of_{b}', ext))2728# evens29ext_evens = frozenset(x for x in range(1, 21) if x % 2 == 0)30hypotheses.append(('rule', 'evens', ext_evens))3132# odds33ext_odds = frozenset(x for x in range(1, 21) if x % 2 != 0)34hypotheses.append(('rule', 'odds', ext_odds))3536# Interval hypotheses: interval_a_b for 1<=a<b<=2037interval_hypotheses = []38for a in range(1, 21):39 for b in range(a+1, 21):40 ext = frozenset(range(a, b+1))41 interval_hypotheses.append(('interval', f'interval_{a}_{b}', ext))4243# Count rules and intervals44rule_hyps = [h for h in hypotheses if h[0] == 'rule']45# Should be 24 total rules46# Let's verify: 11 multiples + 11 powers + 2 = 2447n_rules = len(rule_hyps)48n_intervals = len(interval_hypotheses)4950all_hyps = rule_hyps + interval_hypotheses5152# Prior: 50/50 split between rules and intervals, uniform within each group53# P(h) = 0.5 / n_rules if rule, 0.5 / n_intervals if interval54prior = []55for h in all_hyps:56 if h[0] == 'rule':57 prior.append(0.5 / n_rules)58 else:59 prior.append(0.5 / n_intervals)6061prior = torch.tensor(prior)6263# Examples64examples = [3, 6, 9]6566# Compute likelihood for each hypothesis67log_likelihoods = []68for h in all_hyps:69 ext = h[2]70 # Check if all examples are in the extension71 if all(e in ext for e in examples):72 n = len(ext)73 log_lik = -len(examples) * torch.log(torch.tensor(float(n)))74 log_likelihoods.append(log_lik.item())75 else:76 log_likelihoods.append(float('-inf'))7778log_likelihoods = torch.tensor(log_likelihoods)7980# Compute unnormalized posterior81log_prior = torch.log(prior)82log_posterior_unnorm = log_prior + log_likelihoods8384# Normalize85# Use log-sum-exp for numerical stability86log_Z = torch.logsumexp(log_posterior_unnorm, dim=0)87log_posterior = log_posterior_unnorm - log_Z88posterior = torch.exp(log_posterior)8990# For each query q in 1..20, compute expected probability91# P(q in concept | examples) = sum_h P(h | examples) * I(q in ext(h))92answer = []93for q in range(1, 21):94 prob = 0.095 for i, h in enumerate(all_hyps):96 if q in h[2]:97 prob += posterior[i].item()98 answer.append(prob)99100ANSWER = answer
[0.1406, 0.2989, 1.0000, 0.5153, 0.5153, 1.0000, 0.5153, 0.5153, 1.0000, 0.3843, 0.2934, 0.7125, 0.1789, 0.1414, 0.5968, 0.0886, 0.0696, 0.5387, 0.0410, 0.0093]
[0.1094, 0.2326, 1.0000, 0.4010, 0.4010, 1.0000, 0.4010, 0.4010, 1.0000, 0.2990, 0.2284, 0.7763, 0.1392, 0.1101, 0.6862, 0.0690, 0.0542, 0.6410, 0.0319, 0.0234]
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ✗val- | — | — | · |
Model Sally choosing actions at a vending machine. Actions 'a','b','c' each give 80% chance of one specific food (a→bagel 80%/cookie 10%/doughnut 10%, b→bagel 10%/cookie 80%/doughnut 10%, c→bagel 10%/cookie 10%/doughnut 80%). Sally has a goal food (uniform prior over bagel/cookie/doughnut). If deceptive, she picks an action proportional to P(outcome ≠ goalFood | action); otherwise proportional to P(outcome == goalFood | action). Compute the posterior distribution over Sally's goalFood given that she IS deceptive AND chose action 'b'. Use exact enumeration. pyro, pyro.distributions as dist, and torch are pre-imported. Bind the result to a top-level variable ANSWER as a dict with keys __kind, probs, and support (support sorted alphabetically).
system prompt
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WebPPL primer
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1# Vending machine probabilities2# action 'a': bagel=0.8, cookie=0.1, doughnut=0.13# action 'b': bagel=0.1, cookie=0.8, doughnut=0.14# action 'c': bagel=0.1, cookie=0.1, doughnut=0.856foods = ['bagel', 'cookie', 'doughnut']7actions = ['a', 'b', 'c']89# P(food | action)10vending = {11 'a': {'bagel': 0.8, 'cookie': 0.1, 'doughnut': 0.1},12 'b': {'bagel': 0.1, 'cookie': 0.8, 'doughnut': 0.1},13 'c': {'bagel': 0.1, 'cookie': 0.1, 'doughnut': 0.8},14}1516# For each goalFood, compute P(chose 'b' | deceive=True, goalFood)17# chooseAction(deceive=True): weight of action a = P(outcome != goalFood | action a)18# P('b' selected | deceive, goalFood) = P(outcome!=goalFood|'b') / sum_a P(outcome!=goalFood|a)1920results = {}21for goal in foods:22 weights = {}23 for action in actions:24 p_not_goal = 1.0 - vending[action][goal]25 weights[action] = p_not_goal26 total = sum(weights.values())27 p_b = weights['b'] / total28 results[goal] = p_b2930# Prior over goalFood is uniform (1/3 each)31# Posterior P(goalFood | deceive=True, action='b') ∝ (1/3) * P(action='b' | deceive, goalFood)32# (the deceive=flip() prior: we condition on deceive=True, so that factor cancels)3334unnorm = [results[f] / 3.0 for f in foods]35total_unnorm = sum(unnorm)36norm_probs = [p / total_unnorm for p in unnorm]3738# Sort alphabetically39sorted_foods = sorted(foods)40sorted_probs = [norm_probs[foods.index(f)] for f in sorted_foods]4142ANSWER = {"__kind": "distribution", "probs": sorted_probs, "support": sorted_foods}43
1import torch2import pyro3import pyro.distributions as dist45# Define the vending machine probabilities6# P(food | action)7food_probs = {8 'a': {'bagel': 0.8, 'cookie': 0.1, 'doughnut': 0.1},9 'b': {'bagel': 0.1, 'cookie': 0.8, 'doughnut': 0.1},10 'c': {'bagel': 0.1, 'cookie': 0.1, 'doughnut': 0.8},11}1213foods = ['bagel', 'cookie', 'doughnut']14actions = ['a', 'b', 'c']1516# P(outcome == goalFood | action)17def p_match(action, goal):18 return food_probs[action][goal]1920# P(outcome != goalFood | action)21def p_no_match(action, goal):22 return 1.0 - food_probs[action][goal]2324# If deceptive: picks action proportional to P(outcome != goalFood | action)25# We need P(action='b' | deceptive, goalFood=g)26# = p_no_match('b', g) / sum over actions of p_no_match(a, g)2728def p_action_deceptive(action, goal):29 weights = {a: p_no_match(a, goal) for a in actions}30 total = sum(weights.values())31 return weights[action] / total3233# Posterior over goalFood given deceptive AND action='b'34# P(goalFood=g | deceptive, action='b')35# ∝ P(action='b' | deceptive, goalFood=g) * P(goalFood=g)36# Prior over goalFood is uniform3738log_weights = []39for g in foods:40 p_act = p_action_deceptive('b', g)41 # uniform prior: 1/342 log_weights.append(torch.log(torch.tensor(p_act / 3.0)))4344log_w = torch.tensor(log_weights)45w = log_w.exp()46w = w / w.sum()4748ANSWER = {49 '__kind': 'distribution',50 'probs': w.tolist(),51 'support': sorted(foods) # already alphabetical: bagel, cookie, doughnut52}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 8.9e-9 | 0.0000 | · |
Sally chooses 'a', 'b', or 'c' uniformly at random. She may or may not be deceptive (50/50 flip). If not deceptive, she picks an action whose vending machine outcome matches her goal food; if deceptive, she picks an action whose outcome does NOT match her goal food. The vending machine: 'a' -> bagel 80%, cookie 10%, doughnut 10%; 'b' -> bagel 10%, cookie 80%, doughnut 10%; 'c' -> bagel 10%, cookie 10%, doughnut 80%. Her goal food is uniform over {bagel, cookie, doughnut}.
You observe Sally choose 'b' twice in a row. Using Pyro (with pyro, pyro.distributions as dist, and torch pre-imported), compute the exact posterior over her goal food by manual enumeration. Bind a top-level variable ANSWER to a dict of the form {"__kind": "distribution", "probs": [...], "support": [...]} where support is sorted alphabetically.
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WebPPL primer
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1foods = ['bagel', 'cookie', 'doughnut']2actions = ['a', 'b', 'c']34# Vending machine: P(food | action)5vending = {6 'a': {'bagel': 0.8, 'cookie': 0.1, 'doughnut': 0.1},7 'b': {'bagel': 0.1, 'cookie': 0.8, 'doughnut': 0.1},8 'c': {'bagel': 0.1, 'cookie': 0.1, 'doughnut': 0.8},9}1011def choose_action_prob_b(goal_food, deceive):12 # P(action='b') given goal and deceive13 weights = {}14 for act in actions:15 if deceive:16 # want outcome != goal_food17 weights[act] = 1.0 - vending[act][goal_food]18 else:19 # want outcome == goal_food20 weights[act] = vending[act][goal_food]21 total = sum(weights.values())22 return weights['b'] / total2324# Enumerate over deceive x goalFood25# Prior: deceive=True with p=0.5, goalFood uniform over 3 foods26goal_weights = {'bagel': 0.0, 'cookie': 0.0, 'doughnut': 0.0}2728for deceive in [False, True]:29 p_deceive = 0.530 for goal_food in foods:31 p_goal = 1.0 / 3.032 p_b = choose_action_prob_b(goal_food, deceive)33 # Observe 'b' twice independently34 p_obs = p_b * p_b35 joint = p_deceive * p_goal * p_obs36 goal_weights[goal_food] += joint3738total = sum(goal_weights.values())39norm = {f: goal_weights[f] / total for f in foods}4041sorted_foods = sorted(foods)42probs = [norm[f] for f in sorted_foods]4344ANSWER = {"__kind": "distribution", "probs": probs, "support": sorted_foods}45
1vending = {2 'a': {'bagel': 0.8, 'cookie': 0.1, 'doughnut': 0.1},3 'b': {'bagel': 0.1, 'cookie': 0.8, 'doughnut': 0.1},4 'c': {'bagel': 0.1, 'cookie': 0.1, 'doughnut': 0.8},5}6foods = ['bagel', 'cookie', 'doughnut']7actions = ['a', 'b', 'c']89def action_probs(goal, deceptive):10 if not deceptive:11 weights = [vending[a][goal] for a in actions]12 else:13 weights = [1.0 - vending[a][goal] for a in actions]14 total = sum(weights)15 return {a: w/total for a, w in zip(actions, weights)}1617joint = {}18for food in foods:19 total_prob = 0.020 for deceptive in [False, True]:21 ap = action_probs(food, deceptive)22 total_prob += 0.5 * (ap['b'] ** 2)23 joint[food] = (1.0/3.0) * total_prob2425total = sum(joint.values())26sorted_foods = sorted(foods)27probs_list = [joint[f]/total for f in sorted_foods]2829ANSWER = {30 "__kind": "distribution",31 "probs": probs_list,32 "support": sorted_foods33}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Monty Hall variant in Pyro. Alice picks a door uniformly from {1, 2, 3} (one hides a prize). Monty picks a door uniformly at random from {1, 2, 3} (any door), and we condition on his door being neither Alice's nor the prize door. Build a model that draws aliceDoor and prizeDoor uniformly, samples montyDoor uniformly from {1,2,3}, conditions on montyDoor != prizeDoor and montyDoor != aliceDoor, then returns whether Alice wins under her strategy. Enumerate over all discrete latents manually. pyro, pyro.distributions as dist, and torch are pre-imported. End with a top-level variable ANSWER bound to a dict with keys 'stay' and 'switch', each being a {"__kind": "distribution", "probs": [...], "support": [...]} dict representing P(win) under that strategy. Support should be boolean values [False, True] or [True, False] sorted so that False comes before True. Compute exact probabilities by enumerating all combinations of aliceDoor, prizeDoor, montyDoor.
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WebPPL primer
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1import itertools23doors = [1, 2, 3]45# Enumerate all combinations: aliceDoor, prizeDoor, montyDoor6# Each drawn uniformly from {1,2,3}, weight = 1/27 each7# Condition: montyDoor != prizeDoor AND montyDoor != aliceDoor89stay_win_weight = 0.010stay_total_weight = 0.011switch_win_weight = 0.012switch_total_weight = 0.01314for alice, prize, monty in itertools.product(doors, doors, doors):15 # uniform prior: weight 1/27 for each16 w = 1.0 / 27.017 # condition on monty != prize and monty != alice18 if monty == prize or monty == alice:19 continue20 # stay strategy21 stay_total_weight += w22 if alice == prize:23 stay_win_weight += w24 # switch strategy: alice switches to remaining door25 remaining = [d for d in doors if d != alice and d != monty]26 new_door = remaining[0]27 switch_total_weight += w28 if new_door == prize:29 switch_win_weight += w3031stay_p_win = stay_win_weight / stay_total_weight32stay_p_lose = 1.0 - stay_p_win3334switch_p_win = switch_win_weight / switch_total_weight35switch_p_lose = 1.0 - switch_p_win3637ANSWER = {38 'stay': {39 '__kind': 'distribution',40 'probs': [stay_p_lose, stay_p_win],41 'support': [False, True]42 },43 'switch': {44 '__kind': 'distribution',45 'probs': [switch_p_lose, switch_p_win],46 'support': [False, True]47 }48}49
1import itertools2import torch3import pyro4import pyro.distributions as dist56# Enumerate all combinations7doors = [1, 2, 3]89# We'll accumulate weights for stay_win and switch_win10total_weight = 0.011stay_win_weight = 0.012switch_win_weight = 0.01314for alice, prize, monty in itertools.product(doors, doors, doors):15 # Each combination has equal prior probability (1/27)16 # Condition on monty != prize and monty != alice17 if monty == prize or monty == alice:18 continue1920 # This is a valid scenario21 weight = 1.0 # uniform prior, all equal22 total_weight += weight2324 # Stay strategy: Alice keeps her door25 if alice == prize:26 stay_win_weight += weight2728 # Switch strategy: Alice switches to the remaining door29 # The remaining door is the one that is neither alice nor monty30 remaining = [d for d in doors if d != alice and d != monty]31 # remaining should have exactly one door32 switch_door = remaining[0]33 if switch_door == prize:34 switch_win_weight += weight3536# Normalize37p_stay_win = stay_win_weight / total_weight38p_switch_win = switch_win_weight / total_weight3940# Build distribution dicts41# Support: [False, True] (False=lose, True=win)42stay_dist = {43 "__kind": "distribution",44 "probs": [1.0 - p_stay_win, p_stay_win],45 "support": [False, True]46}4748switch_dist = {49 "__kind": "distribution",50 "probs": [1.0 - p_switch_win, p_switch_win],51 "support": [False, True]52}5354ANSWER = {55 "stay": stay_dist,56 "switch": switch_dist57}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Implement the Monty Hall problem in Pyro. Alice picks a door uniformly from {1, 2, 3}. The prize is behind one door chosen uniformly. Monty opens a door that is neither Alice's nor the prize door (Monty avoids both). Enumerate all combinations of (aliceDoor, prizeDoor, montyDoor) over {1,2,3}^3, keeping only those where montyDoor != aliceDoor and montyDoor != prizeDoor. For the 'stay' strategy, Alice wins if aliceDoor == prizeDoor. For the 'switch' strategy, Alice switches to the remaining unopened door (not aliceDoor, not montyDoor) and wins if that door == prizeDoor. Compute exact posterior probabilities via enumeration.
Bind a top-level variable ANSWER to a dict with keys 'stay' and 'switch'. Each value must be a dict {"__kind": "distribution", "probs": [...], "support": [...]}. For 'stay', the support must be [True, False] with probs [1/3, 2/3]. For 'switch', the support must be [False, True] with probs [1/3, 2/3].
pyro, pyro.distributions as dist, and torch are pre-imported.
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WebPPL primer
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1import itertools23doors = [1, 2, 3]45# Enumerate all valid (aliceDoor, prizeDoor, montyDoor) combinations6stay_counts = {True: 0, False: 0}7switch_counts = {True: 0, False: 0}89for aliceDoor, prizeDoor, montyDoor in itertools.product(doors, doors, doors):10 if montyDoor == aliceDoor or montyDoor == prizeDoor:11 continue12 # Each combination has equal prior weight (1/3 * 1/3 * 1/3 for uniform draws)13 # But montyDoor is sampled from a conditional distribution:14 # montyAvoidBoth gives uniform over valid doors for Monty15 # Weight = P(aliceDoor) * P(prizeDoor) * P(montyDoor | aliceDoor, prizeDoor)16 # P(aliceDoor) = 1/3, P(prizeDoor) = 1/317 # P(montyDoor | aliceDoor, prizeDoor) = uniform over doors != aliceDoor and != prizeDoor18 valid_monty = [d for d in doors if d != aliceDoor and d != prizeDoor]19 monty_prob = 1.0 / len(valid_monty) if montyDoor in valid_monty else 0.020 weight = (1/3) * (1/3) * monty_prob2122 # Stay strategy23 stay_win = (aliceDoor == prizeDoor)24 stay_counts[stay_win] += weight2526 # Switch strategy: switch to remaining door27 remaining = [d for d in doors if d != aliceDoor and d != montyDoor]28 new_door = remaining[0]29 switch_win = (new_door == prizeDoor)30 switch_counts[switch_win] += weight3132# Normalize33stay_total = stay_counts[True] + stay_counts[False]34switch_total = switch_counts[True] + switch_counts[False]3536stay_true_prob = stay_counts[True] / stay_total37stay_false_prob = stay_counts[False] / stay_total3839switch_false_prob = switch_counts[False] / switch_total40switch_true_prob = switch_counts[True] / switch_total4142ANSWER = {43 'stay': {44 '__kind': 'distribution',45 'probs': [stay_true_prob, stay_false_prob],46 'support': [True, False]47 },48 'switch': {49 '__kind': 'distribution',50 'probs': [switch_false_prob, switch_true_prob],51 'support': [False, True]52 }53}54
1import itertools23# Enumerate all combinations of (aliceDoor, prizeDoor, montyDoor) over {1,2,3}^34# Keep only valid ones where montyDoor != aliceDoor and montyDoor != prizeDoor56valid = []7for alice, prize, monty in itertools.product(range(1, 4), repeat=3):8 if monty != alice and monty != prize:9 valid.append((alice, prize, monty))1011# Each valid combination is equally weighted (uniform prior over all three doors)12# Total weight = number of valid combinations1314# For 'stay': Alice wins if aliceDoor == prizeDoor15stay_win = sum(1 for alice, prize, monty in valid if alice == prize)16stay_lose = sum(1 for alice, prize, monty in valid if alice != prize)17total = len(valid)1819stay_win_prob = stay_win / total20stay_lose_prob = stay_lose / total2122# For 'switch': Alice switches to the remaining door (not alice, not monty)23# The remaining door is the one in {1,2,3} that is neither alice nor monty24def remaining_door(alice, monty):25 for d in range(1, 4):26 if d != alice and d != monty:27 return d2829switch_win = 030switch_lose = 031for alice, prize, monty in valid:32 switched = remaining_door(alice, monty)33 if switched == prize:34 switch_win += 135 else:36 switch_lose += 13738switch_win_prob = switch_win / total39switch_lose_prob = switch_lose / total4041# Build the answer dict42# For 'stay': support [True, False], probs [1/3, 2/3]43# For 'switch': support [False, True], probs [1/3, 2/3]4445ANSWER = {46 "stay": {47 "__kind": "distribution",48 "probs": [stay_win_prob, stay_lose_prob],49 "support": [True, False]50 },51 "switch": {52 "__kind": "distribution",53 "probs": [switch_lose_prob, switch_win_prob],54 "support": [False, True]55 }56}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◐TV<.5 | 0.167 | 0.0589 | · |
Monty Hall variant in Pyro. Alice picks a door uniformly from {1, 2, 3}, the prize is behind one door uniformly. Monty picks uniformly from doors that are not Alice's door (he may reveal the prize). We condition on Monty's door being neither Alice's door nor the prize door. Enumerate all (aliceDoor, prizeDoor, montyDoor) combinations manually.
For the 'stay' strategy: Alice wins if aliceDoor == prizeDoor. For the 'switch' strategy: Alice switches to the remaining door (not aliceDoor, not montyDoor), and wins if that door == prizeDoor.
Compute the posterior P(win | strategy) by enumerating all valid combinations. pyro, pyro.distributions as dist, and torch are pre-imported.
Bind a top-level variable ANSWER as a dict with keys 'stay' and 'switch', each being a distribution dict of the form {"__kind": "distribution", "probs": [...], "support": [...]} where support is [False, True] (sorted booleans) with corresponding probabilities. The support must be sorted: False before True.
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WebPPL primer
(primer loads here)
1import itertools23doors = [1, 2, 3]45# Enumerate all (aliceDoor, prizeDoor, montyDoor)6# Prior: uniform over alice, prize, monty7# montyAvoidAlice: monty picks uniformly from doors != alice (prob 1/2 each)8# Condition on: montyDoor != prizeDoor AND montyDoor != aliceDoor910def compute_posterior(switches):11 total_weight = 0.012 win_weight = 0.013 for alice in doors:14 for prize in doors:15 # monty picks from doors != alice, uniformly16 valid_monty_choices = [d for d in doors if d != alice]17 for monty in valid_monty_choices:18 # weight: 1/3 * 1/3 * 1/219 w = (1.0/3) * (1.0/3) * (1.0/len(valid_monty_choices))20 # condition on monty != prize and monty != alice21 if monty == prize or monty == alice:22 continue23 # determine alice's final door24 if switches:25 remaining = [d for d in doors if d != alice and d != monty]26 final_door = remaining[0]27 else:28 final_door = alice29 wins = (final_door == prize)30 total_weight += w31 if wins:32 win_weight += w33 p_win = win_weight / total_weight34 return p_win3536p_win_stay = compute_posterior(False)37p_win_switch = compute_posterior(True)3839ANSWER = {40 'stay': {41 '__kind': 'distribution',42 'probs': [1.0 - p_win_stay, p_win_stay],43 'support': [False, True]44 },45 'switch': {46 '__kind': 'distribution',47 'probs': [1.0 - p_win_switch, p_win_switch],48 'support': [False, True]49 }50}51
1import itertools23# Enumerate all (aliceDoor, prizeDoor, montyDoor) combinations4# Alice picks uniformly from {1,2,3}, prize uniformly from {1,2,3}5# Monty picks uniformly from doors != aliceDoor6# Condition: montyDoor != aliceDoor AND montyDoor != prizeDoor78doors = [1, 2, 3]910stay_wins = 0.011stay_total = 0.012switch_wins = 0.013switch_total = 0.01415for aliceDoor in doors:16 for prizeDoor in doors:17 # Monty picks uniformly from doors != aliceDoor18 monty_choices = [d for d in doors if d != aliceDoor]19 for montyDoor in monty_choices:20 # Prior probability of this combination21 # P(alice) = 1/3, P(prize) = 1/3, P(monty | alice) = 1/222 prior = (1/3) * (1/3) * (1/len(monty_choices))2324 # Condition: montyDoor != aliceDoor (already satisfied) AND montyDoor != prizeDoor25 if montyDoor == prizeDoor:26 continue2728 # Valid combination29 stay_total += prior30 switch_total += prior3132 # Stay strategy33 if aliceDoor == prizeDoor:34 stay_wins += prior3536 # Switch strategy: switch to remaining door37 remaining = [d for d in doors if d != aliceDoor and d != montyDoor]38 # There should be exactly one remaining door39 switchDoor = remaining[0]40 if switchDoor == prizeDoor:41 switch_wins += prior4243# Normalize44stay_win_prob = stay_wins / stay_total45stay_lose_prob = 1.0 - stay_win_prob4647switch_win_prob = switch_wins / switch_total48switch_lose_prob = 1.0 - switch_win_prob4950ANSWER = {51 "stay": {52 "__kind": "distribution",53 "probs": [stay_lose_prob, stay_win_prob],54 "support": [False, True]55 },56 "switch": {57 "__kind": "distribution",58 "probs": [switch_lose_prob, switch_win_prob],59 "support": [False, True]60 }61}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ●TV=0 | 0.000 | 0.0000 | · |
Monty Hall variant in Pyro. Alice picks a door uniformly from {1, 2, 3}. Build a model that enumerates all (aliceDoor, prizeDoor, montyDoor) combinations. Monty's door is sampled uniformly from doors that are not the prize door (he may pick Alice's door). Then condition on montyDoor != prizeDoor AND montyDoor != aliceDoor. For the 'stay' strategy, Alice keeps her original door; for 'switch', she switches to the remaining unopened door. Compute the posterior probability that Alice wins (her door == prize door) under each strategy by manual enumeration over the discrete latent space. pyro, pyro.distributions as dist, and torch are pre-imported. Bind a top-level variable ANSWER as a dict with keys 'stay' and 'switch', each being a dict with __kind, probs, and support fields (support=[False, True] for stay and [False, True] for switch, sorted booleans).
system prompt
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WebPPL primer
(primer loads here)
1doors = [1, 2, 3]23# Enumerate all valid (alice, prize, monty) combinations4# montyAvoidPrize: monty uniform over doors != prize5# condition: monty != alice67stay_win = 0.08stay_lose = 0.09switch_win = 0.010switch_lose = 0.01112for alice in doors:13 for prize in doors:14 # monty is uniform over doors != prize15 valid_monty = [d for d in doors if d != prize]16 monty_prob = 1.0 / len(valid_monty) # = 0.517 for monty in valid_monty:18 # condition on monty != alice19 if monty == alice:20 continue21 # weight = (1/3) * (1/3) * monty_prob22 weight = (1.0/3) * (1.0/3) * monty_prob23 # stay: alice keeps her door24 stay_wins = (alice == prize)25 if stay_wins:26 stay_win += weight27 else:28 stay_lose += weight29 # switch: alice switches to remaining door30 remaining = [d for d in doors if d != alice and d != monty]31 new_door = remaining[0]32 switch_wins = (new_door == prize)33 if switch_wins:34 switch_win += weight35 else:36 switch_lose += weight3738stay_total = stay_win + stay_lose39switch_total = switch_win + switch_lose4041stay_p_win = stay_win / stay_total42stay_p_lose = stay_lose / stay_total43switch_p_win = switch_win / switch_total44switch_p_lose = switch_lose / switch_total4546ANSWER = {47 'stay': {48 '__kind': 'distribution',49 'probs': [stay_p_lose, stay_p_win],50 'support': [False, True]51 },52 'switch': {53 '__kind': 'distribution',54 'probs': [switch_p_lose, switch_p_win],55 'support': [False, True]56 }57}58
1import itertools23# Enumerate all (aliceDoor, prizeDoor, montyDoor) combinations4# Alice picks uniformly from {1,2,3}5# Prize is uniformly from {1,2,3}6# Monty picks uniformly from doors != prizeDoor (he may pick Alice's door)7# Condition on montyDoor != prizeDoor AND montyDoor != aliceDoor89doors = [1, 2, 3]1011stay_wins = 0.012stay_total = 0.013switch_wins = 0.014switch_total = 0.01516for aliceDoor in doors:17 for prizeDoor in doors:18 for montyDoor in doors:19 # Monty picks uniformly from doors != prizeDoor20 monty_choices = [d for d in doors if d != prizeDoor]21 if montyDoor not in monty_choices:22 continue23 monty_prob = 1.0 / len(monty_choices)2425 # Prior probability of this combination26 # P(alice) = 1/3, P(prize) = 1/3, P(monty | prize) = monty_prob27 prior = (1.0/3) * (1.0/3) * monty_prob2829 # Condition on montyDoor != prizeDoor (already enforced above)30 # AND montyDoor != aliceDoor31 if montyDoor == aliceDoor:32 continue3334 # This combination passes the conditioning35 # Stay strategy: Alice keeps aliceDoor36 stay_win = (aliceDoor == prizeDoor)3738 # Switch strategy: Alice switches to the remaining door39 # Remaining door = the door that is not aliceDoor and not montyDoor40 remaining = [d for d in doors if d != aliceDoor and d != montyDoor]41 # There should be exactly one remaining door42 switchDoor = remaining[0]43 switch_win = (switchDoor == prizeDoor)4445 stay_total += prior46 stay_wins += prior * (1.0 if stay_win else 0.0)47 switch_total += prior48 switch_wins += prior * (1.0 if switch_win else 0.0)4950stay_prob_win = stay_wins / stay_total51switch_prob_win = switch_wins / switch_total5253stay_prob_lose = 1.0 - stay_prob_win54switch_prob_lose = 1.0 - switch_prob_win5556ANSWER = {57 'stay': {58 '__kind': 'distribution',59 'probs': [stay_prob_lose, stay_prob_win],60 'support': [False, True]61 },62 'switch': {63 '__kind': 'distribution',64 'probs': [switch_prob_lose, switch_prob_win],65 'support': [False, True]66 }67}
| run | flags | bucket | TV | KL | actions |
|---|---|---|---|---|---|
| sonnet-46-primer-pyro-probmods primary | sonnet-4.6 +primer | ◉TV<.05 | 2.8e-17 | -0.0000 | · |