A Simple Convergence Diagnostic for Markov Chain Monte Carlo

E. Gutiérrez-Peña and J.M. González-Barrios

IIMAS-UNAM, Mexico

Abstract. Markov Chain Monte Carlo techniques draw samples from a multivariate density p( t_1 , . . . , t_d ) by simulating a Markov chain whose equilibrium distribution is precisely p( t_1 , . . . , t_d ). However, although convergence is generally well understood theoretically, diagnosis of convergence is often a difficult task. In practice there are two sides to convergence. First we would like to find all areas of non-negligible probability. Secondly, we would like to obtain enough samples to accurately estimate quantities of interest. It has been argued that the assessment of convergence should be relative to what we wish to achieve by the analysis. Hence most of the convergence diagnostics proposed in the literature focus on the distribution of some real-valued function r = g( t_1 , . . . , t_d ). It must be pointed out, however, that in general one is trying to assess the convergence of all d random variables to the joint distribution and so care must be taken when only one dimensional summaries are examined. In this paper we propose a new convergence diagnostic based on a multivariate nonparametric two-sample test.

Key words: Bayesian inference; convergence of stochastic processes; Gibbs sampler; Metropolis-Hastings algorithm.