A categorical account of the Metropolis-Hastings algorithm

2026-01-30

AI Summary

This work formalizes the core properties of the Metropolis–Hastings (MH) algorithm within a categorical probabilistic framework. By introducing an involution structure in Markov categories and enriching CD categories with commutative monoid structures, it provides the first purely categorical formulation of MCMC methods, unifying the treatment of sub-stochastic kernels, σ-finite measures, and Lebesgue decompositions. The main contribution is a synthetic necessary and sufficient condition for MH reversibility and invariance, yielding a general criterion to verify reversibility with respect to a target distribution. This significantly extends the expressive power and applicability of categorical methods in probabilistic computation and Bayesian inference.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Metropolis-Hastings (MH) is a foundational Markov chain Monte Carlo (MCMC) algorithm. In this paper, we ask 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 CD categories. To go further, we then study 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.

Problem

• Metropolis-Hastings
• categorical probability
• Markov categories
• reversibility
• MCMC

Innovation

• categorical probability
• Metropolis-Hastings algorithm
• Markov categories
• CD categories
• reversibility