💬 NLP

Using Probabilistic Programs to Train Inductive Reasoning in Large Language Models

This paper introduces Program-based Posterior Training (PPT), a novel fine-tuning method that leverages probabilistic programs to generate diverse open-world scenarios and distributional soft labels, thereby significantly enhancing large language models' ability to perform uncertain inductive reasoning and align with human judgments.

Original authors: Liyi Zhang, Akshay K. Jagadish, Brenden M. Lake, Thomas L. Griffiths

Published 2026-06-10

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CC BY 4.0

Original authors: Liyi Zhang, Akshay K. Jagadish, Brenden M. Lake, Thomas L. Griffiths

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a very smart, well-read student (a Large Language Model, or LLM) how to make guesses about the real world.

Currently, these students are excellent at deductive reasoning. This is like solving a math problem or writing code: there is a clear set of rules, and if you follow them correctly, there is one single, 100% correct answer. If the student gets it wrong, it's a clear error.

However, the real world is rarely that neat. Most of our daily thinking is inductive reasoning. This is like being a detective looking at a few scattered clues to figure out what happened.

Example: You see a friend running late, looking tired, and checking their watch. You guess they overslept. But maybe they were stuck in traffic? Maybe they missed the bus? There isn't one "correct" answer; there is a range of possibilities, each with a different level of likelihood.

The problem is that standard training methods struggle with this. It's hard to find a massive library of "detective cases" where the answers are already graded by a teacher who knows the exact probability of every outcome. Asking other AI models to grade these cases is risky because they might just copy their own biases.

The Solution: The "Simulation Game" (Program-based Posterior Training)

The authors of this paper came up with a clever way to train the AI to handle uncertainty. They call it Program-based Posterior Training (PPT).

Think of it like a three-step video game production pipeline:

The Storyteller (The LLM): First, they ask a powerful AI to invent a bunch of wild, open-world scenarios.
- Example: "Imagine a powerlifting championship where athletes' performance depends on their natural strength, their focus that day, and how much they slept."
The Mathematician (The Probabilistic Program): Next, they ask the AI to translate that story into a strict mathematical code (a "probabilistic program"). This code acts like a perfect simulation engine. It doesn't just guess; it calculates the exact mathematical odds of every outcome based on the rules of the story.
- Analogy: If the story is a recipe, this step is writing a computer program that simulates exactly how the cake will rise, how dense it will be, and how likely it is to burn, based on the ingredients.
The Teacher (The Training Data): Finally, they run that simulation thousands of times. Instead of giving the student a single answer (e.g., "Jamal will win"), the simulation gives a distribution of answers (e.g., "Jamal has a 60% chance of winning, a 30% chance of tying, and a 10% chance of losing").

The AI student is then trained on these millions of simulated scenarios. Instead of learning to memorize one "right" answer, it learns to match the shape of the probability cloud generated by the simulation.

What Happened When They Tried It?

The researchers tested this new "student" on several challenges:

Better Guessing: When asked to predict outcomes in sports or healthcare scenarios it had never seen before, the AI made much more accurate estimates. It didn't just guess a number; it understood the uncertainty around that number.
Thinking Like Humans: When they compared the AI's guesses to actual human judgments, the AI trained with this method aligned much better with how real people think. It stopped being overly confident and started acting more like a cautious human observer.
Generalizing: Even when they tested the AI on completely different topics (like switching from sports to healthcare), it still performed well. This suggests it learned a general skill for "thinking with uncertainty," not just memorizing sports facts.
Calibration: The AI became "calibrated." This means if it says there is a 70% chance of something happening, it actually happens about 70% of the time. The paper notes that this improvement is deep; it's not just a surface-level trick (like adjusting a dial after the fact) but a fundamental change in how the model represents uncertainty.

The Bottom Line

The paper argues that to make AI better at real-world reasoning, we shouldn't just feed it more facts. Instead, we should teach it to simulate the world. By using AI to generate stories, turning those stories into strict math simulations, and then training other AIs on the results of those simulations, we can create models that are much better at handling the messy, uncertain, and probabilistic nature of real life.

They didn't test this on medical diagnoses or legal advice in this specific study; they focused on sports, general scenarios, and specific benchmarks to prove the method works. The core takeaway is that probabilistic programs act as a perfect teacher, allowing AI to learn the art of uncertainty.

Technical Summary: Program-based Posterior Training for Inductive Reasoning in LLMs

Problem Statement

While Large Language Models (LLMs) have achieved significant success in deductive reasoning tasks (e.g., mathematics, coding) where correctness is deterministic and verifiable, they struggle with inductive reasoning. Real-world inductive problems require inferring uncertain latent beliefs from sparse, ambiguous, and noisy observations.

Training LLMs for these tasks faces two primary challenges:

Data Scarcity: High-quality, large-scale labeled datasets for open-world inductive reasoning are difficult to curate. Human judgments are expensive and hard to scale, while using stronger LLMs to generate labels risks inheriting their biases and errors.
Distributional Targets: Inductive reasoning often requires representing uncertainty (a distribution over possible answers) rather than a single point estimate. Standard fine-tuning methods struggle with targets that are inherently probabilistic.

Methodology: Program-based Posterior Training (PPT)

The authors propose Program-based Posterior Training (PPT), a pipeline that uses probabilistic programs to generate scalable, grounded supervision for fine-tuning LLMs. The workflow consists of three sequential steps (illustrated in Figure 1):

Scenario Synthesis: A strong closed-source LLM generates diverse, open-world natural language scenarios and associated queries.
Probabilistic Program Generation: The LLM translates each scenario into a structured probabilistic program (specifically in Pyro, a Python-based probabilistic programming language). This step involves:
- Identifying relevant latent variables ( $z$ ), observations ( $x$ ), and query variables ( $y$ ).
- Encoding the generative model $p_\theta(z, x, y) = p_\theta(z)p_\theta(x|z)p_\theta(y|z, x)$ .
Target Generation: The generated probabilistic programs are executed to perform probabilistic inference (via MCMC or Rejection Sampling). This produces the posterior distribution $p_\theta(y|x)$ , which serves as the "ground truth" supervision target.

Fine-Tuning Strategy:
The authors fine-tune open-source LLMs (LLaMA-3-8B-Instruct and Qwen-2-7B-Instruct) using Low-Rank Adaptation (LoRA) on the generated $(scenario, query, posterior)$ tuples. They explore two supervision formats:

Distributional Target (PPT Distribution): The LLM is trained to predict the discretized posterior probability over answer tokens (e.g., integers 0–100) using cross-entropy loss against the program-inferred distribution.
Point Target (PPT Mean): The LLM is trained only on the posterior mean (rounded to the nearest integer).

The dataset comprises over 10,000 unique scenarios and 59,000+ queries across three domains: Sports, Healthcare, and General.

Key Contributions

Novel Training Paradigm: The paper introduces a method to fine-tune LLMs on inductive reasoning by bridging open-world natural language with precise probabilistic models, moving beyond deterministic point-estimate supervision.
Scalable Dataset Creation: The authors create a large-scale dataset of inductive inference problems with probabilistically grounded labels, generated via an automated pipeline extending the Model Synthesis Architecture (MSA).
Empirical Validation of Distributional Supervision: The study demonstrates that fine-tuning on full posterior distributions yields superior results compared to fine-tuning on posterior means or directly using stronger LLMs as labelers.
Complementarity with Calibration Methods: The work shows that PPT improves "raw" calibration (the model's internal uncertainty representation) and is complementary to post-hoc temperature scaling, rather than being subsumed by it.

Experimental Results

1. Inductive Reasoning on Held-Out Scenarios

Accuracy: PPT-fine-tuned models significantly outperform base LLMs and strong closed-source baselines (e.g., Gemini-3.1-Pro) on held-out sports scenarios with unseen inferential motifs. The Mean Absolute Error (MAE) on posterior mean estimates was substantially reduced.
Uncertainty Representation: Unlike base models that often produce uncalibrated, idiosyncratic probability masses (e.g., clustering on multiples of ten), PPT models produce answer distributions that closely align with the ground-truth probabilistic program inferences.
Human Alignment: On sports scenarios, PPT models achieved a correlation ( $R^2$ ) of 0.775 with human judgments, surpassing the base LLM (0.509) and matching the performance of the original MSA programs (0.769).

2. Domain Transfer (Out-of-Domain)

Models fine-tuned on Sports data showed improved performance on Healthcare scenarios (a held-out domain), despite never seeing healthcare training data. This suggests the model learned domain-general inductive behaviors rather than memorizing specific domain facts.
Training on a mix of all domains (Sports, Healthcare, General) provided further improvements over single-domain training.

3. Transfer to External Benchmarks

The authors evaluated transfer on seven external benchmarks, including OpenEstimate (OE), Bayesian Teaching (BT), MMLU, TruthfulQA, HellaSwag, ARC-Challenge, and Winogrande.

Calibration: PPT (Distribution) consistently reduced the Expected Calibration Error (ECE) and Negative Log-Likelihood (NLL) across all multiple-choice benchmarks compared to base models.
Complementarity: Applying post-hoc temperature scaling (TS) to PPT-fine-tuned models yielded further improvements in NLL and ECE, indicating that PPT shifts the model's raw uncertainty representation while TS adjusts the confidence scaling.
Accuracy: PPT improved prior elicitation accuracy (reducing MAE on OpenEstimate) and, in some cases, improved the accuracy of the argmax prediction on multiple-choice tasks.

4. Ablation Studies

Distribution vs. Mean: Fine-tuning on the full distribution (PPT Distribution) consistently outperformed fine-tuning on the posterior mean (PPT Mean) across all metrics, highlighting the importance of distributional information.
Forward Sampling vs. Posterior Inference: While forward sampling allows for larger datasets, models trained on full posterior inference targets achieved better calibration and transfer performance.

Significance and Claims

The paper claims that probabilistic-program-mediated fine-tuning is a promising approach for post-training LLMs to perform approximate inductive inference.

Internalization of Uncertainty: The results suggest that PPT enables models to "deeply internalize" uncertainty, allowing them to generate calibrated beliefs rather than just rescaling outputs post-hoc.
Scalable Supervision: The approach demonstrates that probabilistic programs can serve not only as test-time reasoning tools but also as scalable, automated sources of supervision for training LLMs on complex, uncertain reasoning tasks.
Generalization: The method successfully transfers to unseen inferential structures (motifs) and entirely new domains, suggesting it captures the underlying logic of inductive reasoning rather than surface-level patterns.

The authors acknowledge limitations, noting that the quality of supervision depends on the fidelity of the LLM-generated probabilistic programs and that the current evaluation focuses on numeric and multiple-choice queries rather than open-ended reasoning. However, they conclude that this pipeline effectively improves estimation accuracy, human alignment, and calibration in LLMs.