# Revisiting the Gelman-Rubin Diagnostic

## TL;DR

This paper enhances the Gelman-Rubin convergence diagnostic for MCMC by integrating advanced variance estimators, enabling single-chain calculation, establishing a link to effective sample size, and providing a more stable and principled termination criterion.

## Contribution

The authors upgrade the Gelman-Rubin diagnostic with modern variance estimators, allowing for single-chain use and a new relationship with effective sample size.

## Key findings

- Improved stability in MCMC termination times.
- Diagnostic can now be computed from a single chain.
- Established a direct link between Gelman-Rubin statistic and effective sample size.

## Abstract

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 for estimating variance of Monte Carlo averages. We show that these estimators find immediate use in the Gelman-Rubin statistic, a connection not previously established in the literature. We incorporate these estimators to upgrade both the univariate and multivariate Gelman-Rubin statistics, leading to improved 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 one-to-one relationship between the Gelman-Rubin statistic and effective sample size. Leveraging this relationship, we develop a principled termination criterion for the Gelman-Rubin statistic. Finally, we demonstrate the utility of our improved diagnostic via examples.