Studying Convergence of Markov Chain Monte Carlo Algorithms Using Coupled Sample Paths

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 eeects 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).