Applied Convergence Diagnostics for the Gibbs Sampler

One of the most difficult aspects of using the Gibbs sampler in practice is knowing when to stop the algorithm. In order to answer this we need to have some method which will tell us when we have completed enough iterations for the chain to have converged sufficiently. In this paper I will look at some of the methods that have been suggested in the literature. Most of these methods require input from the user throughout the length of the chain. This aspect of the diagnostics extends the length of time that it takes for the algorithm to terminate and is quite tedious for the user. Ideally one would like to have an automatic algorithm which would test for convergence and stop the Gibbs sampler when it is sufficiently close to convergence. I will look at some of the issues involved in finding such a diagnostic. 1 Introduction Markov Chain Monte Carlo (MCMC) methods have recently become very popular tools for the analysis of Bayesian posterior distributions of relatively high dimension. T..