How Many Iterations in the Gibbs Sampler?

Abstract

Abstract : When the Gibbs sampler is used to estimate posterior distributions (Gelfand and Smith, 1990) the question of how many iterations are required is central to its implementation. When interest focuses on quantiles of functionals of the posterior distribution, we describe an easily-implemented method for determining the total number of iterations required, and also the number of initial iterations that should be discarded to allow for burn-in. The method uses only the Gibbs iterates themselves, and does not, for example, require external specification of characteristics of the posterior density. Here the method is described for the situation where one long run is generated, but it can also be easily applied if there are several runs from different starting points. It also applies more generally to Markov chain Monte Carlo schemes other than the quantities of interest are probabilities rather than full posterior distributions, and when the draws from the posterior distribution are required to be approximately independent. The method is applied to several different posterior distributions. These include a multivariate normal posterior distribution with independent parameters, a bimodal distribution, a cigar-shaped multivariate normal distribution in ten dimensions, and a highly complex 190-dimensional posterior distribution arising in spatial statistics. In each case the method appears to give satisfactory results.