Revisiting the Gelman-Rubin Diagnostic (1812.09384v3)

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.

• The paper introduces an improved estimation technique using advanced batch means methods to enhance the stability of the Gelman-Rubin diagnostic.
• The paper establishes a one-to-one relationship between the GR statistic and effective sample size, enabling a more principled convergence threshold.
• The revised diagnostic is validated through numerical examples, demonstrating robust performance across various distributions and real-world Bayesian models.

Revisiting the Gelman-Rubin Diagnostic: Enhancements and Implications

The paper "Revisiting the Gelman-Rubin Diagnostic," authored by Dootika Vats and Christina Knudson, revisits the widely-utilized Gelman-Rubin (GR) diagnostic for assessing convergence in Markov chain Monte Carlo (MCMC) simulations. Since its inception, the GR diagnostic has remained a fundamental tool due to its simplicity and accessibility. Nevertheless, the authors identify significant limitations concerning its reliability, particularly its potential for premature convergence diagnosis. This paper addresses these issues by introducing modifications that enhance the stability and interpretability of the diagnostic.

Key Contributions

The authors propose two primary developments: an improved estimation technique for the GR statistic incorporating advanced variance estimators, and a systematic approach for selecting an appropriate convergence threshold through the effective sample size (ESS).

1. Improved Estimation Technique: The paper replaces the original variance estimation method within the GR statistic with more efficient estimators developed in recent literature. This modification notably enhances the stability of MCMC termination times. Specifically, the paper employs the replicated lugsail batch means estimator, known for its desirable asymptotic properties, to reliably estimate the Monte Carlo variance.
2. ESS-Based Termination Criterion: A novel one-to-one correspondence between the GR statistic and ESS is established, which allows for a more principled determination of convergence thresholds. The traditional cutoff value of 1.1 is argued to be overly conservative, often resulting in premature convergence claims. By leveraging the connection to ESS, the authors propose a new, theoretically motivated termination threshold that ensures a more robust estimation of target quantities.

Methodology

The methodological approach involves improving the GR statistic through robust statistical estimators that account for correlation in MCMC samples. By shifting from the original estimators to those that use batch means, the proposed method minimizes sensitivity to initial chain conditions, stabilizing the determination of convergence. This is crucial in MCMC, where correlated samples can lead to underestimation of variance and, consequently, inaccurate convergence assessment.

Numerical Illustrations

The authors demonstrate the utility of their improved diagnostic through several illustrative examples, including sampling from a $t_5$-distribution, an autoregressive process, and a multimodal distribution. These examples reveal the inadequacies of the current threshold and highlight the proposed method's strength in avoiding false convergence diagnosis. Additionally, a Bayesian logistic regression analysis of the Titanic dataset illustrates the practical implications of implementing a more stable PSRF in real-world data.

Implications and Future Prospects

The enhancements introduced in this work hold significant implications for both theoretical and applied MCMC practices. The improved stability and interpretability of the GR diagnostic are expected to foster more reliable statistical inference in complex models. Additionally, the authors speculate that further research into variance estimators and their integration into convergence diagnostics could yield even more efficacious tools for MCMC analysis.

While the paper primarily addresses the diagnostic's stability concerning convergence determination, the broader implications suggest an advancement in how practitioners can confidently use MCMC. This foundational improvement sets a precedent for similar advancements in other facets of MCMC diagnostics, potentially prompting further exploration and development within the research community.

In conclusion, Vats and Knudson's paper stands as a compelling contribution that enhances the utility of the GR diagnostic. By systematically addressing its inherent limitations and leveraging novel statistical methods, the research offers a robust pathway towards more reliable MCMC practices.