Monitoring Joint Convergence of MCMC Samplers

We present a diagnostic for monitoring convergence of a Markov chain Monte Carlo (MCMC) sampler to its target distribution. In contrast to popular existing methods, we monitor convergence to the joint target distribution directly rather than a select scalar projection. The method uses a simple nonparametric posterior approximation based on a state-space partition obtained by clustering the pooled draws from multiple chains, and convergence is determined when the estimated posterior probabilities of partition elements under each chain are sufficiently similar. This framework applies to a wide variety of problems, and generalizes directly to non-Euclidean state spaces. Our method also provides approximate high-posterior-density regions, and a characterization of differences between nonconverged chains, all with little additional computational burden. We demonstrate this approach on applications to sampling posterior distributions over Rp, graphs, and partitions. Supplementary materials for this article are available online.