Probabilistic programs provide a rigorous programming-language foundation for expressing rich probabilistic models and offer language-level support for solving challenging statistical inference problems. A key theoretical challenge that probabilistic programs inherit from probability theory is conditioning a probability distribution on an observation that has probability zero. Many languages address this challenge by using computational rules for conditioning that may work well for some problems but are statistically unsound for others. This article introduces a principled mathematical approach for conditioning probabilistic programs. We first show that existing techniques based on "soft conditioning" and "neighborhood limits" exhibit semantic issues from the perspective of measure theory and produce inference results that disagree with sound statistical reasoning. By leveraging tools from geometric measure theory, we construct a novel disintegration-based semantics for probabilistic program traces that enables sound conditioning. The theory delivers an explicit Bayes rule representation of the conditional distribution, which is directly compatible with widely used probabilistic inference algorithms that require access to unnormalized posterior densities. Our Bayes rule handles probabilistic models with loops, branches, discrete, continuous, and mixed-type distributions, observations on manifolds, differentiable structure, and additional modeling motifs that arise in practice.

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