Assessing Convergence and Mixing of MCMC Implementations via Stratification

Abstract

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.

Key Words:

• Batch-means methods
• Bootstrap
• Convergence diagnostics
• Delta method
• Functional central limit theorem
• Mixing
• Stationarity

ACKNOWLEDGMENTS

We are grateful to Ms. Emily (Lei) Kang for her invaluable help with implementing MCMC code for the Pacific Sea Surface Temperature Model, and the associate editor and two referees for their comments and suggestions. This work was supported in part by the National Science Foundation under grant numbers SES-0437251, DMS-0605041, and ATM-0724403, and by the Office of Naval Research under grant N00014-07-1-0512.