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 uses coupled chains from the same sampler, obtained by using the same sequence of random deviates for each run. By examining the distribution of the iteration at which all sample paths couple, convergence properties for the system can be established. 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.