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More accurate behavioral predictions with hybrid Bayesian-connectionist models

Brenden M. Lake, Akshay K. Jagadish, Guangyuan Jiang

arXiv:2608.22154v1cs.LG

TL;DR

The paper introduces BBT as a hybrid approach combining Bayesian modeling with behavioral tuning. BBT produced more accurate behavioral predictions than either component alone and revealed psychologically meaningful departures from the original Bayesian accounts.

  • Problem

    Standard Bayesian modeling does not achieve the behavioral prediction accuracy targeted by this work.

  • Method

    BBT introduces a hybrid approach in which a model is behaviorally tuned using an early-stopping criterion to best predict human behavior.

  • Results

    BBT produced more accurate behavioral predictions than either component alone and revealed psychologically meaningful departures from the original Bayesian accounts.

  • Takeaways & Limitations

    BBT offers improved predictions and psychologically meaningful analyses of how human behavior departs from Bayesian accounts.

  • Takeaways & Limitations

    Interpreting a fine-tuned network requires choices, and the model may miss new hypotheses or changes to existing hypotheses.

Abstract

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Researchers must often choose between Bayesian or neural network models of behavior, two paradigms with complementary strengths and weaknesses. An ideal paradigm would facilitate testing many kinds of representations and inductive biases; Bayesian models make this easy, while neural networks do not. Similarly, an ideal paradigm would avoid over-simplifications; neural networks make this easy, while Bayesian models do not. Here, we introduce Bayesian distillation with Behavioral Tuning (BBT) as an approach to getting the best of both traditions. BBT offers a simple recipe for model building: first, a neural network is trained to mimic a Bayesian model through synthetic data, and second, the network is fine-tuned on human behavior to capture additional structure and nuance. Across four case studies in human concept learning, we find that BBT outperforms traditional approaches at predicting human behavior while also revealing psychological insights, resulting in models that can both mimic Bayesian priors and capture heuristics and biases that violate simple modeling assumptions.

Modeling paradigm

BBT combines Bayesian specification with neural-network approximation and behavioral fine-tuning. The network first learns Bayesian posterior predictions from synthetic episodes, then adapts to human behavior without remaining constrained by the original Bayesian assumptions.

  • Bayesian specification: BBT begins with a Bayesian model whose priors encode task knowledge or hypothesized human inductive biases.
  • Bayesian specification: Bayesian posterior predictive distributions may require an intractable sum over hypotheses, motivating a neural approximation computed with a forward pass.
  • Distillation: During distillation, a network is pretrained on synthetic episodes generated by sampling hypotheses from the prior and input-output data from the likelihood.
  • Distillation: At test time, the frozen network predicts outputs for new queries through forward propagation rather than weight updates or posterior sampling.
  • Behavioral fine-tuning: Behavioral fine-tuning updates the distilled network using human responses to predict how participants answer new inputs.
  • Behavioral fine-tuning: Unlike rigid Bayesian inference, fine-tuning can alter priors, hypothesis-data relations, or add hypotheses that defy straightforward Bayesian assumptions.

Results

Across the evaluated tasks, BBT produced the strongest fits to human behavior while retaining Bayesian structure and incorporating simpler hypotheses, heuristics, and behavioral nuances through fine-tuning.

  • BBT provided the best fit to held-out human responses, outperforming the original Bayesian, distillation-only, and fine-tuning-only models.In one evaluation, it led by 3,175 natural log points; fine-tuning alone performed worst, highlighting the role of distillation.
  • BBT achieved r = 0.81 and RMSE = 0.153, compared with the Bayesian model’s r = 0.66 and RMSE = 0.302.It also outperformed the distillation-only network, which reached r = 0.65 and RMSE = 0.307.
  • Logical concept learning: In logical concept learning, BBT improved behavioral fit while blurring Boolean versus non-Boolean representations and expanding simpler hypotheses.The model’s best-fitting hypotheses used 2.2 fewer function calls on average.
  • Psychological interpretation: BBT remained behaviorally meaningful even when the distillation network diverged from the ground-truth rule, capturing people’s endorsement of simpler “blue” hypotheses and uncertainty over quantifiers.These effects were reflected in both model predictions and internal representations.
  • Shepard concept learning: BBT reproduced human learning trajectories and their interaction with age in the Shepard task when conditioned on block and age group.Participants showed the predicted difficulty ordering Type I < II ≈ III ≈ IV ≈ V < VI, and BBT reproduced the learning pattern across blocks.

Discussion

BBT combines Bayesian distillation with behavioral fine-tuning to improve prediction while retaining interpretable links to Bayesian representations. Across four case studies, it captured psychologically meaningful departures from those original models, though interpretation depends on modeling choices and may miss entirely new hypotheses.

  • Approach: BBT combines Bayesian and connectionist modeling by first distilling a Bayesian model into a neural network, then fine-tuning it on human behavioral data.The recipe is intended to combine structured inductive biases with neural flexibility.
  • Data efficiency: BBT can predict human behavior more accurately than neural networks trained only on behavioral data, especially when human datasets are limited.Distillation and synthetic data helped large networks fit poorly when the initial Bayesian stage was omitted.
  • Interpretability: BBT models need not remain opaque because comparing pre- and post-fine-tuning behavior and representations can identify how they depart from Bayesian models.The approach also supports interpreting how human behavior deviates from ideal Bayesian representations and behavior.
  • Psychological insights: Fine-tuning revealed psychologically meaningful departures from Bayesian accounts, including stronger influence of simpler rules and weaker influence of complex rules.These shifts appeared across numerical, logical, and compositional learning case studies.
  • Limitations: Interpretation is limited because analyses may be restricted to the original Bayesian hypotheses and can miss new hypotheses or altered inference logic.Additional choices about episodes, layers, variables, and likelihoods also affect interpretation and performance, with some benefits remaining speculative.
  • Results: Across four domains, combining Bayesian modeling and connectionist fine-tuning produced more accurate behavioral predictions than either component alone.The studies span learning from very limited data and include concept-learning domains such as logical and compositional rules.

M.1 Implementation of BBT across the case studies

BBT uses a two-stage sequence-to-sequence transformer pipeline: distillation on synthetic Bayesian-model episodes followed by fine-tuning on human behavioral data.

  • Distillation: Distillation minimizes token-averaged cross-entropy between network predictions and Bayesian-model targets across task-specific sampled episodes.Training and validation episode counts vary by case study, including up to 1,000,000 training episodes for Shepard concepts.
  • Two-stage training: BBT first distills Bayesian-model predictions from sampled synthetic episodes, then fine-tunes the resulting network on human behavioral data.The two stages use synthetic targets during distillation and human-response targets during fine-tuning.
  • Architecture: The implementation uses an encoder-decoder transformer with about 30 million trainable parameters, comprising separate encoder and decoder networks.The architecture supports bidirectional attention over concatenated input variables, while the authors describe this choice as an implementation detail.
  • Task-specific batching: Task-specific batching varies the number of episodes, study examples, and queries across number, logical, Shepard, and compositional-instruction learning.For example, distillation uses k = 50 episodes and nq = 20 queries for number concepts, whereas Shepard learning uses k = 100 and nq = 8.
  • Fine-tuning: Fine-tuning mirrors distillation but replaces synthetic targets with examples of human behavior from episodes containing one query and responses from multiple participants.The fine-tuning procedure tracks validation loss and saves the parameter values with the best validation performance.

M.2 Sparse approximation

The sparse approximation procedure interprets BBT predictions as mixtures of human-readable hypotheses, connecting neural outputs to Bayesian posterior predictive structure.

  • Approximation: Sparse approximation identifies hypotheses whose weighted predictions reproduce the network’s behavior across a varied set of queries.The fitted weights are positive, sum to 1, and are encouraged to be sparse for interpretability.
  • Bayesian interpretation: The approximation mirrors a Bayesian posterior predictive, where each hypothesis contributes according to its posterior probability when making the queried prediction.For deterministic hypotheses, the predictive probability is the sum of posterior probabilities for hypotheses predicting 1.
  • Noise and hypotheses: The lapse rate γ absorbs noise while the fitted mixture captures structured predictions, such as number-game responses combining multiples-of-8, ending-in-8, and even hypotheses.For the study set {8, 80, 48}, the example weights are 0.26, 0.24, and 0.17, respectively.
  • Optimization: A Shannon-entropy regularizer concentrates weight on as few hypotheses as possible, with the tradeoff controlled by η = 0.005.The optimization alternates between fitting hypothesis weights and the lapse rate using L-BFGS after removing duplicate predictions.
  • Validation: The authors validated that sparse approximation could recover ground-truth hypotheses from a distilled network’s behavior.This validation establishes that the interpretive procedure can identify the hypotheses underlying synthetic Bayesian-model predictions.

M.3 Visualizing and recovering hypotheses with sparse approximation

Across synthetic concept-learning episodes, sparse approximation recovered generating hypotheses from distilled networks, with accuracy depending on the hypothesis family and task.

  • Number concept learning: For number concepts, exact recovery was evaluated separately for mathematical and interval hypotheses because interval hypotheses are densely distributed.The original Bayesian model was filtered to ensure it could recover the relevant category or exact generating hypothesis under MAP inference.
  • Number concept learning: 97.1% of number-game episodes produced by mathematical hypotheses were recovered, compared with 1.9% of interval hypotheses recovered exactly.A hypothesis counted as recovered when it received the largest fitted weight; exact interval recovery was expected to be difficult.
  • Evaluation design: The case studies used synthetic episodes spanning mathematical, logical, and interval hypothesis families, with task-specific support and query examples.The logical evaluation included 792 episodes, while Shepard episodes varied feature mappings, category labels, and support-example order.
  • Logical concept learning: 99.2% accuracy was achieved when recovering ground-truth logical hypotheses from the distillation-only network while treating other hypotheses as distractors.The evaluation considered the finite hypothesis set shown in Fig. 4D rather than the entire unbounded hypothesis space.
  • Shepard category learning: The Shepard visualization colors episodes by their most likely recovered hypothesis, using distinct colors for Types I–IV and gray for other hypotheses.BBT predictions were conditioned separately on young and older age embeddings, whereas the distillation-only network had no block or age embeddings.

M.4 Fitting the lapse rate

The lapse-rate mechanism converts model token predictions into participant-level output probabilities by mixing model predictions with uniform noise.

  • Known-length outputs: For a known-length output, the lapse mechanism assigns probability P(s) to each possible output symbol s using the model prediction and uniform probability over S.The set S contains the possible output symbols and has cardinality |S|.
  • Variable-length outputs: For variable-length outputs, the same lapse model includes the end-of-sequence token <EOS> among the possible tokens.If the model has no prediction for a symbol, that symbol receives uniform probability 1/|S|.

M.5 Notation for number concept learning

The number-game Bayesian model treats observed numbers as sampled from hypotheses, unlike the other case studies, which condition on observed objects and model only their labels. The neural network approximates the resulting posterior predictive distribution without requiring separate accommodations for these sampling assumptions.

  • In the number game, each observed number x_i is sampled from a hypothesis, producing a likelihood P(x_i|h).
  • In the other case studies, objects are generated by an unmodeled process, and only their labels are sampled conditional on x_i.
  • The model’s posterior predictive distribution and neural approximation therefore differ from the formulation used elsewhere.
  • No architectural adaptation is needed to accommodate the different sampling assumptions across case studies.
  • The network can learn appropriate sampling assumptions through distillation, improving predictions when observations arise from informative sampling.

M.6 Rational rules model

The Rational Rules model represents concepts with hypotheses generated from a DNF grammar and assigns them a frequency-based simplicity prior. Bayesian inference is approximated by Monte Carlo sampling, extension collapsing, and posterior-predictive smoothing with a fitted lapse rate.

  • The model draws 106 hypotheses from a rational-rules prior over a DNF grammar defined on three binary features and eight stimuli.
  • Production rules remain fixed, while production probabilities are independently drawn from a symmetric Dirichlet distribution with α = 1.
  • The grammar excludes the trivial always-true concept by requiring at least one conjunction, and Monte Carlo sampling approximates the posterior.
  • Hypotheses with identical extensions are collapsed, with each prior set to its relative frequency among the 106 samples.
  • Formula frequency implements simplicity bias, while query predictions use a smoothed posterior predictive and a validation-selected lapse rate.
  • The likelihood assigns 1 − ϵ/2 to hypotheses consistent with observed labels and ϵ/2 otherwise, with b = 2 or 4.5 determining ϵ.

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

During fine-tuning for Shepard category learning, the network uses participant-specific trial histories while adding learned block and age embeddings. These embeddings provide information unavailable from within-block support data alone and condition predictions on block–age combinations.

  • BBT therefore augments the network with two learned memory embeddings for experimental block and participant age during fine-tuning.
  • Fine-tuning episodes are constructed separately for each participant, category structure, block, and trial.
  • Each support set contains 1–15 previously completed trials within the block, and the query is the participant’s subsequent response.
  • Because support data contain only trial-level within-block information, the network cannot infer the participant’s block or age from observations alone.
  • Block and age vectors are added after encoding and broadcast across positions, so every support representation carries both signals before decoding.
  • The same fine-tuned network can be conditioned on any block–age combination at inference time.

M.8 Compositional instruction learning

Compositional instruction learning models linguistic input–output mappings with interpretation grammars whose rewrite rules are applied recursively. The study uses a structured curriculum and human data to evaluate generalization to novel and multi-rule queries.

  • An interpretation grammar is a set of rewrite rules for translating linguistic expressions into output sequences.
  • Interpretation proceeds recursively until expressions contain only output symbols and no double brackets.
  • In the worked example, the rule for “dax fep” rewrites the expression into three copies of “dax” before final interpretation.
  • The prior samples grammars with four primitive rules, three function rules, and one concatenation rule from finite input and output symbol sets.
  • Function-rule templates and right-hand-side strings are sampled under constraints on variables, length, termination, and concatenation.
  • Participants learned through four study phases followed by tests requiring generalization to novel queries and, finally, multiple function rules.

Humans

The human concept-learning analyses examine logical concepts, Shepard category learning across age groups, and compositional instruction learning. The figures compare human and model behavior through learning curves, hypothesis analyses, and task episodes.

  • Shepard category learning: Shepard category learning compares BBT with distillation-only models across young and older participants while analyzing changes in inferred hypotheses.The analyses condition BBT representations on age-specific embeddings and use sparse approximation to categorize episodes by their most likely hypothesis.
  • Shepard category learning: Learning curves show human, distillation-only, and BBT performance across task blocks for young and older participants.The four Shepard category structures, Type-I through Type-IV, are plotted as distinct lines, with the y-axis showing the proportion of correct choices.
  • Compositional instruction learning: Compositional instruction learning uses study and test episodes in which pseudowords map linguistic instructions to output sequences.The episodes are generated by a grammar of rewrite rules, and participants and models receive study instructions before producing outputs for query instructions.
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