Geometric Ergodicity of Gibbs Samplers

Due to a demand for reliable methods for exploring intractable probability distributions, the popularity of Markov chain Monte Carlo (MCMC) techniques continues to grow. In any MCMC analysis, the convergence rate of the associated Markov chain is of practical and theoretical importance. A geometrically ergodic chain converges to its target distribution at a geometric rate. In this dissertation, we establish verifiable conditions under which geometric ergodicity is guaranteed for Gibbs samplers in a general model setting. Further, we show that geometric ergodicity of the deterministic scan Gibbs sampler ensures geometric ergodicity of the Gibbs sampler under alternative scanning strategies. As an illustration, we consider Gibbs sampling for a popular Bayesian version of the general linear mixed model. In addition to ensuring the rapid convergence required for useful simulation, geometric ergodicity is a key sufficient condition for the existence of central limit theorems and consistent estimators of Monte Carlo standard errors. Thus our results allow practitioners to be as confident in inference drawn from Gibbs samplers as they would be in inference drawn from random samples from the target distribution.