Performance of the Gibbs, Hit-and-Run, and Metropolis Samplers

Abstract

Abstract We consider the performance of three Monte Carlo Markov-chain samplers—the Gibbs sampler, which cycles through coordinate directions; the Hit-and-Run (HR and the Metropolis sampler, which moves with a probability that is a ratio of likelihoods. We obtain several analytical results. We provide a sufficient condition of the geometric convergence on a bounded region S for the H&R sampler. For a general region S, we review the Schervish and Carlin sufficient geometric convergence condition for the Gibbs sampler. We show that for a multivariate normal distribution this Gibbs sufficient condition holds and for a bivariate normal distribution the Gibbs marginal sample paths are each an AR(1) process, and we obtain the standard errors of sample means and sample variances, which we later use to verify empirical Monte Carlo results. We empirically compare the Gibbs and H&R samplers on bivariate normal examples. For zero correlation, the Gibbs sampler provid...