Calibration, Not Compilation: Detecting and Repairing Misspecified Probabilistic Programs Written by Language Models

2026-06-30

AI Summary

This work addresses the challenge that probabilistic programs generated by language models often suffer from statistical misspecifications—such as incorrect likelihoods, priors, or parameterizations—that are difficult to detect with conventional unit tests. The paper introduces, for the first time, Bayesian calibration as a central criterion for assessing the correctness of probabilistic programs and proposes a fully unsupervised, reference-free framework for their detection and repair. By integrating Bayesian validation techniques—including posterior predictive checks, simulation-based calibration (SBC), sampling diagnostics (e.g., $\hat{R}$, divergences, effective sample size), and held-out predictive log density—the method generates feedback signals to drive an iterative repair loop within large language models. Evaluated on 200 instances, the approach achieves detection AUCs of 0.97 with reference programs and 62–78% without, substantially outperforming unit testing; repair success rates reach 92% and 100% using GPT-5.1 and Claude, respectively.

Abstract

Language models increasingly write probabilistic programs (in NumPyro, Stan, or Pyro), but a program that compiles, runs, and passes every unit test can still be \emph{statistically} wrong -- a Gaussian likelihood for heavy-tailed data, a Poisson for over-dispersed counts, an invalid prior support, or a pathological parameterization. The right verifier is therefore not a test suite but the Bayesian workflow itself: posterior predictive checks, simulation-based calibration, sampler diagnostics ($\hat R$, divergences, ESS), and held-out predictive density. We study this calibration oracle along three axes. \textbf{Detection:} on a benchmark of $14$ misspecification types across $10$ model families ($200$ instances), it flags the bug with AUC $0.97$ ($88\%$ at $2\%$ FPR \emph{when handed the correct reference program, an upper bound}) -- and a fully \emph{reference-free} version that uses no correct program reaches $62$--$78\%$ (the upper figure from a small automated model search), versus $0\%$ for a unit-test oracle. \textbf{Repair:} used as feedback in an LLM repair loop across fifteen models, calibration significantly outperforms unit-test feedback -- which is itself \emph{significantly worse than no feedback at all}, a passing test inducing false confidence that suppresses repair -- and improves over no feedback on strong-but-unsaturated models (GPT-5.1 $33{\to}92\%$, Claude $75{\to}100\%$; paired McNemar, $n{=}228$). \textbf{Reality:} on programs LLMs write from scratch for neutral briefs, $15$--$47\%$ of runnable ones are statistically misspecified (unit tests catch none), and calibration-guided repair significantly beats LLM-as-judge review, a Bayesian-workflow checklist, and data-summary self-debug. Across all three, the lesson is the same: for probabilistic programs, correctness is calibration, not compilation.

Problem

• probabilistic programs
• statistical misspecification
• calibration
• language models
• Bayesian workflow

Innovation

• calibration
• probabilistic programming
• model misspecification
• Bayesian workflow
• LLM repair