Assessing Convergence and Mixing of MCMC Implementations via Stratification

Some posterior distributions lead to Markov chain Monte Carlo (MCMC) chains that are naturally viewed as collections of subchains. Examples include mixture models, regime-switching models, and hidden Markov models. We obtain MCMC-based estimators of posterior expectations by combining different subgroup (subchain) estimators using stratification and poststratification methods. Variance estimates of the limiting distributions of such estimators are developed. Based on these variance estimates, we propose a test statistic to aid in the assessment of convergence and mixing of chains. We compare our diagnostic with other commonly used methods. The approach is illustrated in two examples: a latent variable model for arsenic concentration in public water systems in Arizona and a Bayesian hierarchical model for Pacific sea surface temperatures. Supplementary materials, which include MATLAB codes for the proposed method, are available online.