A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler conver...

. I describe a simple procedure for investigating the convergence properties of Markov Chain Monte Carlo sampling schemes. The procedure employs multiple runs from a sampler, using the same random deviates for each run. When the sample paths from all sequences converge, it is argued that approximate equilibrium conditions hold. The procedure also provides a simple diagnostic for detecting modes in multimodal posteriors. Several examples of the procedure are provided. In Ising models, the relation between the correlation parameter and the convergence rate of rudimentary Gibbs samplers is investigated. In another example, the effects of multiple modes on the convergence of coupled paths are explored using mixtures of bivariate normal distributions. The technique is also used to evaluate the convergence properties of a Gibbs sampling scheme applied to a model for rat growth rates (Gelfand et al 1990). Acknowledgements I would like to thank Steve MacEachern, Julian Besag, Donald Rubin, A...

this paper, we propose an extension of the Gibbs coupling algorithm that eliminates each of these difficulties. The proposed method for studying the convergence properties of general MCMC algorithms may be summarized as follows. For a specified MCMC algorithm, c chains are started from initial values drawn at random from an over-dispersed estimate of the stationary distribution. Each of the chains is updated according to its MCMC conditional distributions, except that updates are made jointly in a way that allows each pair of the c chains to couple at a random time. The iteration at which the c chains couple is recorded, and the process is repeated m times. Based on the m coupling times, the quantiles of the distribution of the coupling iteration are estimated and used to obtain a bound on the total variation distance of the MCMC iterates from the stationary distribution of the chain. Joint updates of the c chains are made either by a generalization of maximal coupling to multiple chains, or an approximation to this c chain Markovmaximal coupling obtained through simple mixture sampling. Like the coupling procedure proposed by Johnson (1996), the algorithm described above has strong connections to more theoretical studies of coupling in Markov chains and MCMC algorithms. Particularly relevant works in this direction include Doeblin (1933), Griffeath (1975), Pitman (1976), Goldstein (1979), Lindvall (1992), Meyn and Tweedie (1993), and Rosenthal (1995). Related work in the more general area of MCMC convergence diagnostics are reviewed in Besag et al 1995, and include Frigessi et al (1992), Gelman and Rubin (1992), Geyer (1992), Roberts (1992), Besag and Green (1993), Lund and Tweedie (1993), Smith and Roberts (1993), Garren and Smith (1994), Gelman et al (1994), and Robe...