On the Geometric Ergodicity of Two-Variable Gibbs Samplers

TL;DR

This paper investigates the conditions under which two-variable Gibbs samplers exhibit geometric ergodicity, providing criteria for both deterministic and random scan versions, and characterizing their convergence rates.

Contribution

It introduces a sufficient condition for geometric ergodicity of both Gibbs sampler versions and a method to establish subgeometric ergodicity, advancing understanding of their convergence behavior.

Findings

Both versions are geometrically ergodic under the given condition.

The paper characterizes convergence rates for a specific family of discrete bivariate distributions.

Provides a unified approach to analyze ergodicity of Gibbs samplers.

Abstract

A Markov chain is geometrically ergodic if it converges to its in- variant distribution at a geometric rate in total variation norm. We study geo- metric ergodicity of deterministic and random scan versions of the two-variable Gibbs sampler. We give a sufficient condition which simultaneously guarantees both versions are geometrically ergodic. We also develop a method for simul- taneously establishing that both versions are subgeometrically ergodic. These general results allow us to characterize the convergence rate of two-variable Gibbs samplers in a particular family of discrete bivariate distributions.