A categorical account of the Metropolis-Hastings algorithm

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.

• The paper introduces a categorical framework that abstracts the MH algorithm's invariance and reversibility through enriched CD categories.
• It recasts classical measure-theoretic notions, such as Radon–Nikodym derivatives and Lebesgue decompositions, into algebraic diagrammatic reasoning.
• The approach unifies standard, auxiliary-variable, and nonreversible MH variants, offering templates for correct and extendable MCMC design.

A Categorical Account of the Metropolis-Hastings Algorithm

Introduction and Motivation

The Metropolis-Hastings (MH) algorithm is a central component of Markov chain Monte Carlo (MCMC) methodology, underpinning a vast array of algorithms for sampling from complex probability distributions. Despite its extensive use in statistics and machine learning, formal abstractions that capture the essential correctness properties and generalizability of MH-type kernels have been limited largely by measure-theoretic frameworks. This paper, "A categorical account of the Metropolis-Hastings algorithm" (2601.22911), investigates whether categorical probability theory—primarily through Markov and copy-discard (CD) categories—can provide both a synthetic and general perspective on MH and its modern extensions, particularly in the context of involutive frameworks.

The work leverages recent categorical probability formalisms to recast notions such as invariance, reversibility, and augmentation, all fundamental to MCMC algorithms, within categorical structures. Most importantly, it introduces CD categories enriched over commutative monoids as a minimal yet expressive framework for reasoning about summation, decomposition, and essential measure-theoretic properties like absolute continuity and singularity—thereby providing a formal basis for analyzing both classical and non-standard MH procedures.

Categorical Probability Frameworks: Markov and CD Categories

The categorical approach begins with Markov categories, a class of symmetric monoidal categories modeling compositional probabilistic reasoning with morphisms representing Markov kernels. Fundamental concepts in MCMC, such as invariance and reversibility, are given succinct categorical definitions:

• Invariance: $P$ is $\mu$-invariant if $P \circ \mu = \mu$.
• Reversibility: A categorical analogue of detailed balance, whereby $P$ is $\mu$-reversible if the corresponding diagram is invariant under swap.

Notably, complex structures such as state space augmentation—critical for advanced MCMC (e.g., auxiliary variable and pseudo-marginal methods)—are captured by categorical composition involving Bayesian inverses. This allows a unified treatment of augmentation techniques using only abstract categorical properties, removing algorithm-specific measure-theoretic intricacies.

However, MH kernels in practice often involve unnormalized or non-stochastic components, necessitating a shift from Markov categories to the broader CD categories, which support arbitrary composition of substochastic kernels via the manipulation of "effects". The effect structure allows reasoning about unnormalized weights (e.g., Radon–Nikodym derivatives) needed for handling proposals and acceptance functions in MH-type algorithms.

Semiadditive Enrichment and Algebraic Decomposition

To fully encapsulate practical MH algorithms, which involve convex combinations (e.g., sums of transition kernels for accept/reject steps), the paper studies CD categories enriched over commutative monoids (semiadditive CD categories). This innovation allows:

• Addition of morphisms: Essential for expressing full MH kernels as sums of accept and reject transitions.
• Convex combinations and reweighting: Formalizing mixtures and importance weighting algebraically.
• Decomposition and absolute continuity: Expressing measure-theoretic decompositions (e.g., Lebesgue) categorically.

Within this framework, substochastic kernels, finite and $\sigma$-finite measures, absolute continuity, singularity, and Lebesgue decomposition are all realized as categorical constructs. The enrichment yields a preorder on morphisms, generalizes Radon-Nikodym derivatives as effects, and enables pointwise measure-theoretic reasoning as diagrammatic categorical manipulations.

Main Theoretical Results

The principal technical development is a categorical derivation of the necessary and sufficient conditions for MH kernel reversibility, paralleling Theorem 3 of [Andrieu et al. 2020]. Given a measure $\mu$ and an involution $\phi$, the categorical formulation identifies a measurable subset $S$ such that reversibility decomposes according to equivalence and singularity of $\mu$ and its pushforward:

Balancing Condition: The full MH kernel is reversible with respect to $\mu$ if and only if the acceptance probability $\alpha$ satisfies

$$\alpha(\xi) = \alpha(\phi(\xi)) \, r(\xi) \quad \text{for } \xi \in S$$

where $r$ is the Radon–Nikodym derivative of the pushforward of $\mu$ restricted to $S$ under $\phi$.

Key claims and contributions include:

• Recovery of standard detailed balance as a special case.
• Synthetic derivation of balancing conditions for reversibility via string diagrams—removing reliance on measure-theoretic calculations.
• Direct generalization to skew-reversible and nonreversible MH-type samplers by categorical manipulations.
• Extension to abstract Lebesgue decomposition in enriched CD categories, leading to general statements about decomposition into absolutely continuous and singular parts.

These results are established entirely within the categorical formalism, indicating that the algebraic structure is not merely descriptive but also prescriptive for deriving and extending correctness properties of MCMC algorithms.

Implications and Outlook

The implications for theoretical and practical statistics are multifaceted:

• Unification: The categorical approach subsumes classical, auxiliary-variable, and involutive MCMC variants, providing uniform invariance and reversibility proofs.
• Generality: The algebraic/diagrammatic perspective is general enough to include state space augmentations, nonreversible kernels, convex combinations, and importance sampling.
• Algorithm design: The synthetic conditions derived can serve as templates for verifying new or existing algorithms, especially those employing non-standard proposals or extensions like random transformations, mirror couplings, or parameter space augmentation.
• Software and automation: Diagrammatic reasoning aligns naturally with programmable frameworks, facilitating the development of probabilistic programming systems that check or synthesize correct MCMC kernels mechanically [cf. (Cusumano-Towner et al., 2020)].

Furthermore, the enrichment approach prompts deeper investigation into when and how categorical analogues of standard measure-theoretic theorems (e.g., Radon–Nikodym, Lebesgue decomposition) can be established. This connects with foundational work in effectus theory and the formal semantics of probabilistic programming languages.

Conclusion

Through the categorical formalism, this work demonstrates that fundamental correctness properties and architectural features of the Metropolis-Hastings algorithm, including recent involutive and nonreversible extensions, can be derived and generalized in a purely algebraic setting. The minimal enrichment of CD categories over commutative monoids enables reasoning about kernels involving sums, decompositions, and their measure-theoretic analogues, opening new avenues for both theoretical exploration and practical algorithm development in MCMC and stochastic computation. Future directions include categorical treatments of further classical probabilistic theorems and the design of effect-centric probabilistic programming frameworks leveraging these insights.