More accurate behavioral predictions with hybrid Bayesian-connectionist models

Modeling paradigm

• Modelers use the Bayesian toolkit to specify a task’s hypothesis space and prior inductive biases, which determine how hypotheses are favored.For a given set of input-output examples, the model predicts a human response to a new input.Source: Modeling paradigm · pp. 5–6
• The network approximates the generally intractable posterior predictive sum with a forward pass; at test time, weights remain frozen as it processes each new episode.This is in-context, or memory-based, meta-learning rather than adapting weights to each episode through gradient descent.Source: Modeling paradigm · pp. 6–7
• During distillation, the network learns to approximate the Bayesian posterior predictive distribution from synthetic episodes sampled from the prior and likelihood.Generating these episodes does not require posterior probabilities or posterior sampling.Source: Modeling paradigm · pp. 6–7
• BBT distills a Bayesian model into a neural network, then fine-tunes it on human behavior to improve predictions beyond standard Bayesian modeling.Behavioral fine-tuning is not constrained by the original prior, likelihood, or logic of Bayesian inference.Source: Modeling paradigm · p. 7
• When supported by human behavior, fine-tuning can change the prior or how hypotheses relate to data, or add hypotheses that defy straightforward assumptions.Source: Modeling paradigm · p. 7

Number concept learning

• BBT fit human judgments best (Pearson r = 0.81, RMSE = 0.153), versus Bayesian (r = 0.66, RMSE = 0.302) and distillation-only (r = 0.65, RMSE = 0.307).The comparison aggregates across query and support-set combinations.Source: Figure 3 · p. 9, Results · p. 11
• Pearson r = 0.997 (n = 5,100): on held-out sets, the distilled network’s query-endorsement probabilities strongly correlated with the Bayesian model’s.The correlation aggregates across all 51 held-out sets.Source: Figure 3 · p. 9, Results · p. 10
• Networks organized support sets by likely hypothesis class without seeing those labels during training, and BBT’s internal representations differed from distillation-only.After BBT fine-tuning, the influence of “evens”/“odds” and “ending in n” hypotheses expanded.Source: Figure 3 · p. 9, Results · pp. 12–13
• The prior-fit Bayesian model scored -31,419.3 log-likelihood (r = 0.67, RMSE = 0.301), and full-fit scored -28,736.7 (r = 0.78, RMSE = 0.166), still 578.8 natural log points behind BBT.Prior-fit changed the prior; full-fit also allowed a flexible likelihood, with 518,569 parameters.Source: Results · pp. 11–12
• 3,175 log points: BBT led held-out human-response log-likelihood, followed by the Bayesian and distillation-only models; fine-tuning-only performed worst.Models were evaluated on held-out human responses.Source: Results · pp. 10–11, Table 1 · p. 12
• For the program producing {66, 78}, the Bayesian model narrows in on “multiples of 6” (posterior probability 0.53); BBT is consistent with “evens” (0.46, versus Bayesian 0.05).Source: Results · p. 12
• For {8, 80, 48}, Bayesian inference favored “multiples of 8” (0.89), while BBT and humans also endorsed “ending in 8” and “evens”; BBT missed humans’ endorsement of numbers in the 80s.Source: Results · p. 12

Results

• After BBT fine-tuning, evens, odds, and ending-in-n hypothesis classes had much greater influence, reorganizing parts of the embedding previously dominated especially by intervals and powers.The embedding for {66, 78} was in part of the space dominated by evens; {8, 80, 48} was lumped with other classes including ending-in-n.Source: Results · p. 13

Logical concept learning

• BBT best fit held-out judgments, beating the 21 Bayesian models and distillation-only and fine-tuning-only networks by at least 7,826 natural log points.The Bayesian comparison was against the 21 models considered by Piantadosi et al..Source: Results · p. 16
• BBT provided the best fit (r = 0.88, RMSE = 0.190, n = 2,657), compared with distillation-only (r = 0.87, RMSE = 0.214) and fine-tuning-only (r = 0.57, RMSE = 0.325).Fig. 4B and C show the final 8 sets for two example concepts.Source: Results · p. 16
• Across five pre-training runs, fine-tuning improved the pre-trained network by at least 481.7 log points on 11 held-out concepts absent from fine-tuning sequences.This analysis tested generalization to new concepts because the original split used sequences from concepts seen during fine-tuning.Source: Results · p. 16
• Although both models were 69% correct on ground-truth rules, people and BBT partly endorsed a blue object that violated “largest blue,” while BBT captured uncertainty about quantifiers in small sets better than distillation alone.The pattern suggests that the simpler “blue” hypothesis can appeal, and that people are uncertain about how quantifiers apply to small sets.Source: Results · p. 17
• Some people and BBT were tempted by blue or green objects more generally without quantifier constraints, consistent with BBT’s greater emphasis on simpler hypotheses than the distillation-only network (Extended Data Fig. 6).The example is shown in Fig. 4C.Source: Results · p. 17
• In logical-task PCA, distillation-only embeddings separated Boolean from quantifier-based non-Boolean hypotheses, while fine-tuning blurred that division and expanded the influence of simple hypotheses.The figure caption specifies that true, false, and individual-feature hypotheses expand after fine-tuning.Source: Figure 4 · p. 15, Results · p. 17

Earlier case study

• BBT embeddings retained correlations with “yes” probability and entropy (r = 0.986 and r = 0.877) while blurring the Boolean/non-Boolean separation.Before fine-tuning, these components explained 54.4% of embedding variance.Source: Results · pp. 17–18
• BBT shifted toward simpler hypotheses, with best-fitting hypotheses averaging 2.2 fewer function calls than before fine-tuning.The simplest hypotheses expanded at the expense of hypotheses with quantifiers.Source: Results · p. 18

Shepard concept learning

• Participants’ difficulty ordering matched Boolean complexity: Type I < II ≈ III ≈ IV ≈ V < VI.Source: Results · p. 19
• Reported log likelihoods were -5,554.4 for Rational Rules, -5,601.1 for distillation only, and -5,924.7 for fine-tuning only.Source: Results · p. 20
• At block 1, BBT favored Type I single-feature rules more than distillation only, especially for younger participants (0.02 → 0.17 versus 0.02 → 0.05 for older adults).The authors interpret this as participants favoring simpler rules beyond the Rational Rules prior, with younger participants inferring them more quickly.Source: Results · pp. 20, 22
• BBT transferred across Shepard studies: held-out Badham log likelihood was -5,331.1 after Lewandowsky fine-tuning, close to -5,298.7 after Badham fine-tuning and better than fine-tuning only (-5,926.5) or distillation only (-5,565.7).The authors interpret this as evidence that the distilled Rational Rules prior provides a substrate for behavioral fine-tuning to transfer across datasets.Source: Results · p. 22

Results

• Participants matched the task’s rule-based symbolic system on an average of 80.7% of responses, while nearly 20% followed other patterns.The study included 23 participants.Source: Results · p. 23
• BBT’s behavioral fine-tuning gains 122.8 natural log points in human-behavior log-likelihood over the distillation-only network, with a much larger gain over the fine-tuning-only network.The comparison evaluates models’ predictions of human behavior on the instruction-learning task.Source: Results · p. 25
• One-to-one errors made up 24.4% of human errors and 44.4% of BBT errors, versus 10.4% of distillation-only errors.The one-to-one bias maps one input symbol to one output symbol.Source: Results · p. 25
• Iconic-concatenation errors involving function 3 made up 23.3% of human errors and 7.0% of BBT errors, versus 0% of distillation-only errors.This bias maps the first input argument to the first output position, rather than the reverse required by function 3.Source: Results · p. 25
• BBT captured these biases without hand-designing training to teach them, and performed slightly better than the model with hand-engineered biases.The distillation-only network made fewer errors overall, but its errors were less human-like than BBT’s.Source: Results · p. 26

Discussion

• Fine-tuning lets BBT predict human behavior more accurately than mimicry alone, including in domains thought to favor Bayesian models.The authors describe the resulting analyses as psychologically meaningful accounts of how human behavior strays from existing models.Source: Discussion · pp. 27, 29
• In case study 4, a 30-million-parameter model trained on 166 participants’ eight queries each predicted novel participants on a novel task, while fine-tuning-only networks fit poorly.The authors argue that distilling on synthetic data before fine-tuning can enable larger, flexible networks to learn from limited human data.Source: Discussion · p. 27
• Unlike Centaur, BBT uses a controlled training pipeline and trains the network to perform the task, avoiding reliance on prior human responses that may enable shortcuts.This controlled setup lets researchers examine model performance and deviations from ideal Bayesian representations and behavior.Source: Discussion · p. 28
• In case studies 1–3, BBT shifted toward simpler rules: evens, odds, and rules ending in n; rules without quantifiers or constants; and more Type-I hypotheses.Compared with the classic Bayesian model, case study 1 showed weaker influence from powers-of-n and multiples-of-n rules; case study 2 shifted away from quantified rules.Source: Discussion · p. 28
• In case study 4, BBT recovered characteristic biases in compositional instruction learning, consistent with prior work that manually added them to model training.Comparing behavior and internal representations before and after fine-tuning helped identify departures from the initial Bayesian models.Source: Discussion · pp. 28–29
• BBT’s interpretation depends on choices about what to visualize, and analyses restricted to the original Bayesian hypotheses may miss new hypotheses or altered inference logic.The authors suggest that automated interpretation or training networks to show their reasoning could help address this limitation.Source: Discussion · p. 29
• Closing the loop between BBT and automated interpretation could let the system discover symbolic hypotheses and feed them back into the Bayesian model’s hypothesis space.Source: Discussion · p. 30
• BBT’s two training stages are not a developmental claim that children begin near Bayesian solutions and then adapt; this work instead aims to identify models that best predict how people learn new concepts.Source: Discussion · p. 30
• A developmentally oriented alternative would distill plausible innate structure into a network, then train it on realistic egocentric input such as children’s headcam video rather than adult behavior.Source: Discussion · p. 30
• Across four domains, Bayesian modeling with connectionist fine-tuning predicted behavior more accurately than either component alone and revealed psychologically meaningful departures from the original Bayesian accounts.Source: Discussion · p. 30

M.1 Implementation of BBT across the case studies

• Training minimized token-averaged cross-entropy, first against synthetic sequences from the Bayesian model and then against examples of human behavior.The two-stage process is illustrated in Fig. 2.Source: §M.1 · p. 31
• The model was an approximately 30-million-parameter sequence-to-sequence transformer with an encoder for concatenated inputs and a decoder that predicted the output.Encoder-decoder attention supported bidirectional attention over the concatenated input variables; the authors note a decoder-only architecture should also work.Source: §M.1 · p. 31
• Distillation ran for 50 epochs with AdamW, a 0.0001 base learning rate, warm-up followed by validation-loss plateau scheduling, gradient clipping at 1.0, and a halt after the fourth rate reduction.Scheduler patience was four epochs, except five for logical concepts.Source: §M.1 · pp. 31–32
• Distillation datasets sampled 100,000 training and 200 validation episodes for number concepts, 200,000 and 400 for logical concepts, 1,000,000 and 2,000 for Shepard concepts, and 100,000 and 200 for compositional instruction learning.The compositional episodes were the same episodes used for training the MLC.Source: §M.1 · p. 32
• Distillation batches aggregated episodes with task-specific settings: k=50 for number and compositional learning, k=100 for logical and Shepard concepts, with query counts of 20, 10, 10, and 8, respectively.The number of study examples m was sampled from task-specific ranges.Source: §M.1 · p. 32
• After distillation, the network was fine-tuned on human behavior for 20 epochs, or 50 for Shepard learning, and the best-validation-loss weights were saved.Validation loss was tracked during fine-tuning, but training was not actually stopped early.Source: §M.1 · pp. 32–33
• Fine-tuning batches used task-specific episode counts, while each episode paired one input query with responses from different human participants, so response counts varied with participant availability.Episode counts k were 25 for number concepts, 10 for logical concepts, 100 for Shepard concepts, and 10 for compositional instruction learning.Source: §M.1 · p. 33

• Neural-network predictions are approximated as a sparse, interpretable mixture of hypotheses from the original Bayesian model, with positive normalized weights and a lapse rate for noise.Source: §M.2 · pp. 33–34
• For set {8, 80, 48}, BBT predictions can be approximated by assigning weights 0.26 to the multiples-of-8 hypothesis, 0.24 to ending-in-8, and 0.17 to evens.Source: §M.2 · p. 34
• Weights and lapse rate are fitted across varied queries using entropy regularization to concentrate mass on few hypotheses (η=0.005), with L-BFGS alternating parameter fits after equivalent hypotheses are removed.Source: §M.2 · p. 34
• The algorithm recovered ground-truth hypotheses from distilled neural-network behavior.Source: §M.2 · p. 34

Number concept learning

• 97.1% of mathematical-hypothesis episodes, but only 1.9% of interval-hypothesis episodes, had their best-fitting hypotheses recovered exactly from the distillation-only model.Episodes were filtered so the original Bayesian model could recover the exact generating mathematical hypothesis or the correct interval-versus-mathematical category; recovery meant receiving the largest fitted weight.Source: §M.3 · p. 35

Logical concept learning

• 99.2% accuracy was achieved recovering logical-concept ground-truth hypotheses from the distillation-only network, with other Fig. 4D hypotheses as distractors rather than the full unbounded space.The episodes used relatively simple hypotheses across several classes.Source: §M.3 · p. 35

Shepard category learning

• In Shepard category learning, episodes were visualized by their most likely hypothesis inferred through sparse approximation, with Types I–IV assigned distinct colors and other hypotheses gray.BBT predictions used separate young and older age embeddings, each conditioned on block 1; the distillation-only network had no block or age embeddings.Source: §M.3 · p. 36

M.4 Fitting the lapse rate

• For known-length outputs, participant symbol probabilities combine the model prediction before lapsing with a uniform lapse distribution over possible symbols.The possible-symbol set S has cardinality |S|.Source: §M.4 · p. 36
• For outputs of unknown length, the lapse model adds the end-of-sequence token (<EOS>) to the possible-token set.Source: §M.4 · p. 36
• If the model has no prediction for a symbol, such as one beyond its predicted sequence, that symbol receives probability 1/|S|.Source: §M.4 · p. 36

M.5 Notation for number concept learning

• In the number game, each observed number x_i is sampled from a hypothesis, yielding likelihood P(x_i|h); all observed numbers have label 1.Source: §M.5 · p. 36
• In the other case studies, observations x_i come from an unmodeled process and only labels y_i are sampled, yielding P(y_i|h; x_i).The semicolon indicates that the model conditions on x_i rather than modeling its distribution; the posterior predictive and neural approximation therefore differ from Eq. 1.Source: §M.5 · p. 37
• The network needs no adaptation for different sampling assumptions because distillation teaches it to model informative sampling when that improves predictions.Source: §M.5 · p. 37

Rational rules model

• 106 hypotheses were sampled from the Rational Rules prior over a DNF grammar on three binary features and evaluated on all eight stimuli.The setup followed.Source: §M.6 · p. 37
• Production rules were fixed, while production probabilities were independently drawn from a symmetric Dirichlet distribution with α = 1 and marginalized during hypothesis generation.The start rule required at least one conjunction, excluding the always-true concept; Monte Carlo sampling approximated the posterior, with identical extensions collapsed and priors set by frequency among the 106 samples.Source: §M.6 · p. 37
• The model used the outlier likelihood, assigning 1 − ϵ/2 to labels consistent with a hypothesis and ϵ/2 otherwise, with b = 2 or 4.5.The outlier probability was ϵ = e−b/(1 + e−b).Source: §M.6 · p. 37
• Query predictions came from the posterior predictive dataset, were smoothed by the same outlier probability, and were combined with a lapse rate selected by validation grid search over [0, 0.9].The validation split was from the respective behavioral dataset.Source: §M.6 · pp. 37–38

M.7 Learning block and age embeddings for Shepard category learning

• Fine-tuning episodes were constructed separately for each participant, category structure, block, and trial, using completed trials in that block as support and the next response as the query.The support set included between 1 and 15 completed trials.Source: §M.7 · p. 38
• Because within-block support data cannot reveal block or age, the network learns block and age embeddings that are added to encoded support representations before decoder attention.The embeddings are injected after encoding to prevent attenuation by the encoder.Source: §M.7 · p. 38
• The embeddings are jointly optimized with the backbone, while the otherwise unchanged fine-tuned network can be conditioned at inference on any block–age combination.The embedding table contains eight 512-dimensional vectors, about 4,000 parameters.Source: §M.7 · pp. 38–39

Interpretation grammars

• Interpretation grammars map linguistic expressions to output sequences through rewrite rules applied recursively until only output symbols remain.The hypothesis representation is inspired by formal semantics.Source: §M.8 · p. 39

Prior over interpretation grammars

• The prior specifies four primitive rules and three function rules plus concatenation, sampling primitive mappings and variable-based function rules.Function-rule templates take one or two arguments, and their right-hand sides use variables from the rule’s left-hand side.Source: §M.8 · pp. 39–40

Human data collection

• Eight episodes from different grammars, each with 14 study examples and 10 queries excluding catch trials, were selected because distillation-only networks scored 100% correct on queries across all five random seeds.Source: §M.8 · p. 40
• Participants studied three function rules in separate phases, then took a final test requiring multiple rules to answer correctly.The study used instruction and phase quizzes, catch trials, and exclusions; data from 166 participants were used to fine-tune BBT.Source: §M.8 · p. 41

Humans

• The hypothesis analysis compares Shepard episodes between distillation-only and BBT using sparse approximation, age-specific BBT embeddings, and decoder-layer-3 PCA.Episodes are categorized by likely hypothesis and rule literal count; BBT results are separated for young and older participants.Source: Figure 7 · p. 51
• BBT (b = 2) reproduces human Shepard-task difficulty ordering, including older participants’ swap between Type IV and Type II, unlike distillation-only and fine-tuning-only models.The learning curves cover young and older participants and four Shepard category structures; distillation-only shows Type I < Type II < Type III < Type IV on synthetic tasks.Source: Figure 8 · p. 52
• In compositional instruction learning, participants and models map pseudoword instructions to color-circle output sequences generated by rewrite-rule grammars.The figure shows a test episode and two fine-tuning episodes; query outputs are shown but were not shown to participants.Source: Figure 9 · p. 53