Bin Yu

1. STATISTICS WITH DEPENDENT DATA

1.2 Markov Chain Monte Carlo

The Markov Chain Monte Carlo (MCMC) method has attracted much attention for its potential as an important computational tool for a variety of applications including likelihood computation in frequentist statistics and posterior computation in Bayesian statistics. Moreover, it is currently used by many statisticians in their routine data analysis and statistical inference. The MCMC method enables us to obtain (dependent) samples from a target density from which direct sampling is difficult. Quantities of interest for the target distribution, such as mean, variance, and tail probabilities, can then be approximated using the MCMC sample. Since the target distribution is the stationary distribution for the constructed Markov chain, the success of the MCMC method relies crucially on our ability to construct sensible error estimates based on a MCMC sample and to assess the convergence of the chain to its equilibrium. My interest in MCMC has focused on the use of regeneration ideas for output error assessment and convergence diagnostics for MCMC. proposes to use the split-chain idea of Nummelin, Athreya and Ney. In this way, known results for regenerative simulations apply and more reliable estimates of the variance of the sample mean can be obtained from the simulated chain. We also suggest a regeneration plot as a diagnostic tool for the convergence of the chain. The effective use of this proposal, which has been successfully implemented for some known schemes, will depend on finding an efficient split of the Markov chain. In the discussion on the on the Besag et al. paper in Statistical Science we propose the cusum plot as a simple diagnostic tool based on a one-dimensional summary of the MCMC sample. Strong approximation results for absolutely regular sequences are used to argue that the smoothness of the line-joined cusum path reflects the mixing speed of the one-dimensional summary statistic: the faster the mixing, the more ``hairy" the cusum plot. A benchmark path is advised to assess the ``hairiness" of the path. Experimental evidence on MCMC diagnostics gathered in the Ph.D. thesis of Steve Brooks at Cambridge shows that the cusum plot compares very favorably with existing techniques. See also Working with the Metropolis algorithm, a popular form of MCMC, attempts to use the known form of the unnormalized target density and a kernel estimator for convergence diagnostics. This idea was made more practical, in combination with Quasi Monte Carlo methods, in