Geometric Ergodicity & Scanning Strategies For Two-Component Gibbs Samplers

TL;DR

This paper investigates the geometric ergodicity of two-component Gibbs samplers under various scanning strategies, establishing conditions for rapid convergence and comparing different update methods.

Contribution

It proves that geometric ergodicity under one strategy implies it under others, providing verifiable conditions for convergence in Gibbs samplers.

Findings

Geometric ergodicity is equivalent across different scanning strategies.

Provided simple conditions to verify geometric ergodicity.

Illustrated results with two practical examples.

Abstract

In any Markov chain Monte Carlo analysis, rapid convergence of the chain to its target probability distribution is of practical and theoretical importance. A chain that converges at a geometric rate is geometrically ergodic. In this paper, we explore geometric ergodicity for two-component Gibbs samplers which, under a chosen scanning strategy, evolve by combining one-at-a-time updates of the two components. We compare convergence behaviors between and within three such strategies: composition, random sequence scan, and random scan. Our main results are twofold. First, we establish that if the Gibbs sampler is geometrically ergodic under any one of these strategies, so too are the others. Further, we establish a simple and verifiable set of sufficient conditions for the geometric ergodicity of the Gibbs samplers. Our results are illustrated using two examples.