The Performances of Gelman-Rubin and Geweke's Convergence Diagnostics of Monte Carlo Markov Chains in Bayesian Analysis


  • Han Du University of California Author
  • Zijun Ke Department of Psychology, Sun Yat-sen University, China Author
  • Ge Jiang Department of Educational Psychology, University of Illinois Urbana-Champaign, USA Author
  • Sijia Huang Counseling and Educational Psychology Department, Indiana University Bloomington, USA Author



Convergence diagnostics , Bayesian analysis, Gelman-Rubin diagnostic, Geweke diagnostic


Bayesian statistics have been widely used given the development of Markov chain Monte Carlo sampling techniques and the growth of computational power. A major challenge of Bayesian methods that has not yet been fully addressed is how we can appropriately evaluate the convergence of the random samples to the target posterior distributions. In this paper, we focus on Gelman and Rubin's diagnostic (PSRF), Brooks and Gleman's diagnostic (MPSRF), and Geweke's diagnostics, and compare the Type I error rate and Type II error rate of seven convergence criteria: MPSRF>1.1, any upper bound of PSRF is larger than 1.1, more than 5% of the upper bounds of PSRFs are larger than 1.1, any PSRF is larger than 1.1, more than 5% of PSRFs are larger than 1.1, any Geweke test statistic is larger than 1.96 or smaller than -1.96, and more than 5% of Geweke test statistics are larger than 1.96 or smaller than -1.96. Based on the simulation results, we recommend the upper bound of PSRF if we only can choose one diagnostic. When the number of estimated parameters is large, between the diagnostic per parameter (i.e., PSRF) or the multivariate diagnostic (i.e., MPSRF), we recommend the upper bound of PSRF over MPSRF. Additionally, we do not suggest claiming convergence at the analysis level while allowing a small proportion of the parameters to have significant convergence diagnosis results.

Author Biography

  • Han Du, University of California

    Dr. Han Du is assistant professor at UCLA. Her methodological interests have evolved along three inter-related lines: (1) Bayesian methods and statistical computing; (2) longitudinal data analysis and time series analysis; and (3) study design and sample size determination. From a substantive perspective, she am interested in applying quantitative methods in developmental, clinical, cognitive, educational, and health research. Specifically, I focus on developing and applying appropriate Bayesian, longitudinal, and time series models and developing applicable statistical methods for analyzing various complex data and facilitating study design.






Theory and Methods