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

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 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. 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 when handed the correct reference program, an upper bound) -- and a fully 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. Repair: used as feedback in an LLM repair loop across fifteen models, calibration significantly outperforms unit-test feedback -- which is itself 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→92%, Claude 75→100%; paired McNemar, n=228). 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.