Weak Poincar\'{e} inequalities for Deterministic-scan Metropolis-within-Gibbs samplers

TL;DR

This paper uses weak Poincaré inequalities to analyze convergence of deterministic-scan Metropolis-within-Gibbs samplers, providing explicit subgeometric bounds and insights into their behavior for Bayesian inference applications.

Contribution

It introduces novel comparison techniques for convergence analysis of nonreversible Markov chains and establishes fundamental results for weak Poincaré inequalities in discrete-time chains.

Findings

Joint chain inherits marginal chain's convergence properties.

Explicit subgeometric convergence bounds derived.

Applications demonstrated in Bayesian hierarchical and diffusion models.

Abstract

Using the framework of weak Poincar\'{e} inequalities, we analyze the convergence properties of deterministic-scan Metropolis-within-Gibbs samplers, an important class of Markov chain Monte Carlo algorithms. Our analysis applies to nonreversible Markov chains and yields explicit (subgeometric) convergence bounds through novel comparison techniques based on Dirichlet forms. We show that the joint chain inherits the convergence behavior of the marginal chain and conversely. In addition, we establish several fundamental results for weak Poincar\'{e} inequalities for discrete-time Markov chains, such as a tensorization property for independent chains. We apply our theoretical results through applications to algorithms for Bayesian inference for a hierarchical regression model and a diffusion model under discretely-observed data.