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A categorical account of the Metropolis-Hastings algorithm

Rob Cornish, Andi Q. Wang

TL;DR

The paper develops a categorical framework for Metropolis-Hastings by embedding MH into Markov and copy-discard categories and then enriching CD categories over commutative monoids to handle the full MH kernel. It provides a synthetic account of invariance, reversibility, and skew-reversibility, including a purely algebraic derivation of the MH balancing condition and a converse, plus an extension to augmented-state and skew-reversible variants. It also establishes an abstract Lebesgue-decomposition toolkit in this setting and shows how it recovers the classical ML/measure-theoretic results for MH-type algorithms. Overall, the work demonstrates that categorical probability can illuminate and generalize core MCMC analyses, with potential applications to a broader class of MH-type samplers and statistical methods.

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 $σ$-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.

A categorical account of the Metropolis-Hastings algorithm

TL;DR

The paper develops a categorical framework for Metropolis-Hastings by embedding MH into Markov and copy-discard categories and then enriching CD categories over commutative monoids to handle the full MH kernel. It provides a synthetic account of invariance, reversibility, and skew-reversibility, including a purely algebraic derivation of the MH balancing condition and a converse, plus an extension to augmented-state and skew-reversible variants. It also establishes an abstract Lebesgue-decomposition toolkit in this setting and shows how it recovers the classical ML/measure-theoretic results for MH-type algorithms. Overall, the work demonstrates that categorical probability can illuminate and generalize core MCMC analyses, with potential applications to a broader class of MH-type samplers and statistical methods.

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 -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.
Paper Structure (43 sections, 35 theorems, 121 equations, 1 algorithm)

This paper contains 43 sections, 35 theorems, 121 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. MCMC foundations in Markov categories
  3. Recap: Markov categories
  4. Invariance and reversibility
  5. Skew-reversibility
  6. Reversibility via augmented state spaces
  7. The first Metropolis term in CD categories
  8. Recap: CD categories
  9. The first summand of Metropolis--Hastings
  10. Semiadditive CD categories
  11. Basic definitions
  12. Substochasticity, probabilities, and convex combinations
  13. Aside: connection with distributive Markov categories
  14. Cancellative and finite morphisms
  15. The full MH kernel
  16. ...and 28 more sections

Key Result

theorem 1

Suppose $\mu$ is a finite measure on a measurable space $\mathcal{E}$, and $\phi : \mathcal{E} \to \mathcal{E}$ is a measurable involution. Then there exists a measurable $S \subseteq \mathcal{E}$ such that the restriction $\mu|_{S}$ and its pushforward $\mu|_{S}^\phi$ are equivalent as measures, an where $r \coloneqq \frac{\mathrm{d} \mu|_{S}^\phi}{\mathrm{d}\mu|_{S}}$ denotes the Radon--Nikodym

Theorems & Definitions (98)

  • theorem 1
  • definition 1
  • definition 2: Page 9, Cho2019
  • definition 3
  • proposition 1
  • proof
  • proposition 2
  • proof
  • Remark 2
  • definition 4
  • ...and 88 more