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arxiv: 2606.09856 · v1 · pith:KJITCPSEnew · submitted 2026-05-26 · 💻 cs.CL · cs.AI· cs.LG· stat.ML

Using Probabilistic Programs to Train Inductive Reasoning in Large Language Models

Liyi Zhang , Akshay K. Jagadish , Brenden M. Lake , Thomas L. Griffiths This is my paper

Pith reviewed 2026-06-29 18:41 UTC · model grok-4.3

classification 💻 cs.CL cs.AIcs.LGstat.ML
keywords inductive reasoningprobabilistic programslarge language modelsfine-tuningsoft labelsuncertainty calibrationposterior trainingdistributional targets
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The pith

Program-based Posterior Training uses probabilistic programs to generate soft labels that improve LLMs on inductive reasoning tasks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Program-based Posterior Training (PPT) to train large language models on inductive reasoning tasks. Standard fine-tuning struggles with inductive problems because they involve uncertain, distributional targets and lack large labeled datasets. PPT has an LLM generate diverse scenarios as probabilistic programs, performs inference to create soft labels, and fine-tunes the model on those. Evaluations show improved accuracy on held-out tasks, better match to human judgments, and gains on external benchmarks that go beyond simple temperature scaling for calibration.

Core claim

PPT substantially improves estimation accuracy on held-out inductive tasks, increases alignment with human judgments, and transfers to external benchmarks for estimation and calibration. The gains in raw calibration are not subsumed by post-hoc temperature scaling, showing that the models have more deeply internalized uncertainty compared to output rescaling.

What carries the argument

Program-based Posterior Training (PPT), which uses LLMs to generate open-world scenarios as probabilistic programs and runs probabilistic inference to produce distributional target responses for fine-tuning.

If this is right

  • Improved performance on held-out inductive reasoning motifs
  • Increased alignment with human inductive judgments
  • Transfer of gains to external estimation and calibration benchmarks
  • Calibration improvements that persist beyond post-hoc temperature scaling

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hybrid approaches combining symbolic probabilistic programs with neural fine-tuning could address other forms of uncertain reasoning in AI systems.
  • This method might reduce the need for expensive human-labeled data in training for real-world decision making under uncertainty.
  • Further work could test if the approach generalizes to more complex or domain-specific inductive problems beyond the generated scenarios.

Load-bearing premise

The distributional targets produced by the LLM-generated probabilistic programs and inference are faithful representations of real-world inductive reasoning rather than artifacts of the generation process.

What would settle it

If fine-tuning on PPT labels shows no improvement over baselines when evaluated on a set of inductive tasks constructed entirely from human-authored scenarios without LLM generation.

Figures

Figures reproduced from arXiv: 2606.09856 by Akshay K. Jagadish, Brenden M. Lake, Liyi Zhang, Thomas L. Griffiths.

Figure 1
Figure 1. Figure 1: Workflow from data generation to LLM fine-tuning. A sequential prompting process with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effects of Program-based Posterior Training. (a) Example held-out scenario and queries. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mean absolute error (MAE) where colors indicate different models, base and fine-tuned. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Figure reports mean absolute error (MAE) on three evaluation sets: sports training [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Per-query mean response and standard-deviation between humans and LLMs on held-out [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean estimates on Jamal’s strength on eight scenarios that progressively goes from weaker [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
read the original abstract

Post-training Large Language Models (LLMs) for reasoning typically focuses on deductive tasks such as mathematics and coding where correctness is verifiable. Yet, many real-world reasoning problems are inductive: agents must infer uncertain beliefs from sparse, ambiguous observations. There are challenges to using standard fine-tuning methods for inductive reasoning, including difficulties in curating large-scale, high-quality labeled datasets and in handling targets that are inherently distributional. In this work, we introduce a novel approach, called Program-based Posterior Training (PPT), to address these limitations: we use an LLM to generate diverse open-world scenarios as probabilistic programs, run probabilistic inference to produce distributional target responses to queries, and then fine-tune on these probabilistic soft labels. Using this approach, we fine-tune LLMs on 10,000 programmatically generated scenarios and evaluate on held-out motifs, human-labeled judgments, and external benchmarks. Overall, PPT substantially improves estimation accuracy on held-out inductive tasks, increases alignment with human judgments, and transfers to external benchmarks for estimation and calibration. Additionally, the gains in raw calibration are not subsumed by post-hoc temperature scaling, showing that the models have more deeply internalized uncertainty compared to output rescaling. Together, these results suggest that probabilistic-program-mediated fine-tuning is a promising approach for post-training LLMs to reliably perform approximate inductive inference.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces Program-based Posterior Training (PPT): an LLM generates diverse open-world scenarios as probabilistic programs, probabilistic inference produces distributional soft labels for queries, and the LLM is fine-tuned on 10k such labels. Evaluation on held-out motifs, human judgments, and external benchmarks is claimed to show substantial gains in estimation accuracy, human alignment, and calibration that are not explained by post-hoc temperature scaling.

Significance. If the central claims are substantiated, PPT offers a scalable route to distributional training targets for inductive reasoning without manual curation of labels, addressing a key limitation of standard fine-tuning on deductive tasks. The separation of program generation from inference and the demonstration that calibration gains exceed temperature scaling would be notable contributions to post-training methods for uncertainty-aware reasoning.

major comments (2)
  1. [Abstract] Abstract: the claims that PPT 'substantially improves estimation accuracy on held-out inductive tasks' and 'increases alignment with human judgments' are presented without any quantitative results, confidence intervals, statistical tests, or ablation controls. This is load-bearing for the central claim because the abstract supplies the only evidence summary available for assessing whether the data support the reported gains.
  2. [Abstract] Abstract (PPT pipeline description): the method assumes that inference over LLM-generated probabilistic programs yields distributional targets faithful to real-world inductive reasoning rather than artifacts of the generator LLM or program syntax. No validation of program correctness, coverage of ambiguity, or calibration against independent human data is described; if the programs systematically encode the generator's priors, measured improvements could reflect learning of those idiosyncrasies rather than general inductive inference.
minor comments (1)
  1. [Abstract] The abstract refers to 'held-out motifs' and 'external benchmarks' without naming them or citing the specific datasets or prior work they come from.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment point by point below, indicating where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claims that PPT 'substantially improves estimation accuracy on held-out inductive tasks' and 'increases alignment with human judgments' are presented without any quantitative results, confidence intervals, statistical tests, or ablation controls. This is load-bearing for the central claim because the abstract supplies the only evidence summary available for assessing whether the data support the reported gains.

    Authors: We agree that the abstract would be strengthened by including quantitative results. In the revised manuscript we will update the abstract to report specific metrics (e.g., accuracy gains, alignment scores) with confidence intervals and references to the statistical comparisons performed in the main text. revision: yes

  2. Referee: [Abstract] Abstract (PPT pipeline description): the method assumes that inference over LLM-generated probabilistic programs yields distributional targets faithful to real-world inductive reasoning rather than artifacts of the generator LLM or program syntax. No validation of program correctness, coverage of ambiguity, or calibration against independent human data is described; if the programs systematically encode the generator's priors, measured improvements could reflect learning of those idiosyncrasies rather than general inductive inference.

    Authors: The manuscript reports evaluation on held-out motifs and human-labeled judgments, which directly tests generalization and alignment with independent human data. These results provide evidence that performance gains are not limited to artifacts of the generator. We will add a brief discussion of program-generation limitations and any additional checks on program validity in the revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper describes generating 10k scenarios as probabilistic programs via LLM, running inference to obtain distributional soft labels, and fine-tuning on those labels, with evaluation on held-out motifs, human judgments, and external benchmarks. No equations, definitions, or steps in the abstract reduce any claimed prediction or result to the inputs by construction (e.g., no self-definitional terms, no fitted parameter renamed as prediction, no load-bearing self-citation chain). The probabilistic inference step and independent external evaluations keep the chain self-contained; any dependence on the generator LLM is not exhibited as a formal equivalence in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no equations or detailed methods available to enumerate free parameters or invented entities.

axioms (2)
  • domain assumption LLM can generate diverse open-world scenarios as probabilistic programs that are suitable for inference
    Central to the PPT pipeline described in the abstract.
  • domain assumption Probabilistic inference on those programs yields distributional targets that improve inductive reasoning when used for fine-tuning
    Required for the claim that soft labels transfer to held-out tasks and human judgments.

pith-pipeline@v0.9.1-grok · 5781 in / 1273 out tokens · 28486 ms · 2026-06-29T18:41:01.051035+00:00 · methodology

discussion (0)

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