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Weak Poincaré inequalities for Deterministic-scan Metropolis-within-Gibbs samplers

Mengxi Gao, Gareth O. Roberts, Andi Q. Wang

TL;DR

This work develops a weak Poincaré inequality (WPI) framework for deterministic-scan Metropolis-within-Gibbs (MwG) samplers, enabling subgeometric convergence analysis even in nonreversible settings. By embedding Dirichlet forms and K^*-WPI comparisons, the authors relate the convergence of a two-component MwG chain to its Gibbs and conditional updates, and establish a tensorization property for WPIs on product spaces. They provide explicit transfer mechanisms between joint and marginal chains, and between full MwG and marginal variants, yielding computable rate functions F whose inverse track decay via F^{-1}. The theory is illustrated through a Normal-Inverse-Gamma toy, a Bayesian hierarchical model, and data augmentation for discretely observed diffusion processes (including Ornstein–Uhlenbeck), with clear prescriptions for exponential vs polynomial rates depending on inner-update scaling and problem structure.

Abstract

Using the framework of weak Poincaré 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 Poincaré 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.

Weak Poincaré inequalities for Deterministic-scan Metropolis-within-Gibbs samplers

TL;DR

This work develops a weak Poincaré inequality (WPI) framework for deterministic-scan Metropolis-within-Gibbs (MwG) samplers, enabling subgeometric convergence analysis even in nonreversible settings. By embedding Dirichlet forms and K^*-WPI comparisons, the authors relate the convergence of a two-component MwG chain to its Gibbs and conditional updates, and establish a tensorization property for WPIs on product spaces. They provide explicit transfer mechanisms between joint and marginal chains, and between full MwG and marginal variants, yielding computable rate functions F whose inverse track decay via F^{-1}. The theory is illustrated through a Normal-Inverse-Gamma toy, a Bayesian hierarchical model, and data augmentation for discretely observed diffusion processes (including Ornstein–Uhlenbeck), with clear prescriptions for exponential vs polynomial rates depending on inner-update scaling and problem structure.

Abstract

Using the framework of weak Poincaré 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 Poincaré 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.
Paper Structure (20 sections, 47 theorems, 223 equations, 2 figures, 3 algorithms)

This paper contains 20 sections, 47 theorems, 223 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main result and structure
  4. Related work
  5. Preliminaries
  6. Markov chain theory
  7. Weak Poincaré inequalities
  8. Two-component Gibbs samplers
  9. Comparison results for DG Sampling
  10. MwG sampler with two intractable conditionals
  11. Marginal MwG sampler with one intractable conditional
  12. Tensor products as Gibbs Samplers
  13. Toy Example: Normal Inverse Gamma Distribution
  14. Bayesian Hierarchical Model
  15. Data Augmentation for Diffusion Processes
  16. ...and 5 more sections

Key Result

Theorem 2.1

andrieu2022comparison Suppose a WPI with $\beta: (0, \infty) \rightarrow [0, \infty)$ holds for $\mu$ and $T^*T$. Then, for any $f \in \mathrm{L}^2_0(\mu)$ such that $\|f\|^2_\mathrm{osc} < \infty$ and any $n \in \mathbb{N}$, where $F: \left(0, \frac{1}{4}\right] \rightarrow \mathbb{R}$ is the decreasing convex and invertible function $F(x) :=\int_x^{\frac{1}{4}} \frac{\dif v}{\mathrm{K}^*(v)}$,

Figures (2)

  • Figure 1: Relationship among two-component Gibbs samplers.
  • Figure 2: Relationship among Gibbs samplers and their X-marginal version.

Theorems & Definitions (106)

  • Definition 1: Weak Poincaré inequality
  • Theorem 2.1
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1: andrieu2022comparison
  • Lemma 2.2
  • Theorem 2.2: Chaining Poincaré inequalities
  • Remark 2.3
  • Lemma 2.3
  • ...and 96 more