Denotational validation of higher-order Bayesian inference

TL;DR

This paper develops a modular, denotational semantics framework for higher-order Bayesian inference algorithms in probabilistic programming, overcoming technical challenges with quasi-Borel spaces and validating algorithms like SMC and MCMC.

Contribution

It introduces a novel semantic account using quasi-Borel spaces to model higher-order probabilistic programs and validates common inference algorithms within this framework.

Findings

Validated inference algorithms such as Sequential Monte Carlo and MCMC.

Developed semantic structures supporting higher-order functions and continuous distributions.

Proved a quasi-Borel space version of the Metropolis-Hastings-Green theorem.

Abstract

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.