Measure Transformer Semantics for Bayesian Machine Learning
This work addresses the need for a foundational semantics in probabilistic programming for Bayesian inference, which is incremental as it builds on existing measure theory and inference engines.
The paper tackles the challenge of providing a rigorous semantics for probabilistic programming languages used in Bayesian machine learning, resulting in a core functional calculus that supports discrete, continuous, and hybrid measures, including zero-probability events, and enables efficient approximate inference with thousands of observations per second for realistic models.
The Bayesian approach to machine learning amounts to computing posterior distributions of random variables from a probabilistic model of how the variables are related (that is, a prior distribution) and a set of observations of variables. There is a trend in machine learning towards expressing Bayesian models as probabilistic programs. As a foundation for this kind of programming, we propose a core functional calculus with primitives for sampling prior distributions and observing variables. We define measure-transformer combinators inspired by theorems in measure theory, and use these to give a rigorous semantics to our core calculus. The original features of our semantics include its support for discrete, continuous, and hybrid measures, and, in particular, for observations of zero-probability events. We compile our core language to a small imperative language that is processed by an existing inference engine for factor graphs, which are data structures that enable many efficient inference algorithms. This allows efficient approximate inference of posterior marginal distributions, treating thousands of observations per second for large instances of realistic models.