Gelman and Rubin’s (1992) convergence diagnostic is one of the most popular methods for terminating a Markov chain Monte Carlo (MCMC) sampler. Since the seminal paper, researchers have developed sophisticated methods of variance estimation for Monte Carlo averages.

In “Revisiting the Gelman-Rubin Diagnostic,” we show that this class of estimators finds immediate use in the Gelman-Rubin statistic, a connection not established in the literature before. We incorporate these estimators to upgrade both the univariate and multivariate Gelman-Rubin statistics, leading to increased stability in MCMC termination time. An immediate advantage is that our new Gelman-Rubin statistic can be calculated for a single chain. In addition, we establish a relationship between the Gelman-Rubin statistic and effective sample size. Leveraging this relationship, we develop a principled cutoff criterion for the Gelman-Rubin statistic. Finally, we demonstrate the utility of our improved diagnostic via examples.