Compositional Semantics for Probabilistic Programs with Exact Conditioning

TL;DR

This paper introduces a probabilistic programming language with exact conditioning for Gaussian variables, providing formal semantics and generalizing to broader probabilistic settings using categorical methods.

Contribution

It presents a new language with exact conditioning, formal semantics, and a categorical framework that generalizes conditioning beyond Gaussian distributions.

Findings

Established operational, denotational, and equational semantics.

Proved properties like exchangeability of conditions.

Generalized conditioning using categorical formulations.

Abstract

We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal properties, thus generalizing to wider settings. We define the Cond construction, which internalizes conditioning as a morphism, providing general compositional semantics for probabilistic programming with exact conditioning.