Metropolis-Hastings (MH) is a foundational Markov chain Monte Carlo (MCMC) algorithm. In this talk I will present our recent work (link) that asks whether it is possible to formulate and analyse MH in terms of categorical probability, using a recent involutive framework for MH-type procedures as a concrete case study. We show how basic MCMC concepts such as invariance and reversibility can be formulated in Markov categories, and how one part of the MH kernel can be analysed using standard copy-discard (CD) categories. To go further, we then consider enrichments of CD categories over commutative monoids. This gives an expressive setting for reasoning abstractly about a range of important probabilistic concepts, including substochastic kernels, finite and sigma-finite measures, absolute continuity, singular measures, and Lebesgue decompositions. Using these tools, we give synthetic necessary and sufficient conditions for a general MH-type sampler to be reversible with respect to a given target distribution.
This talk aims to be accessible with minimal background in MCMC or probability theory.