On the use of a local $\hat{R}$ to improve MCMC convergence diagnostic

TL;DR

This paper introduces a localized version of the $ ilde{R}$ convergence diagnostic for MCMC, focusing on quantiles to better detect convergence issues across the distribution.

Contribution

It proposes a new localized $ ilde{R}$ and a novel indicator $ ilde{R}_ ext{infty}$ that improve convergence diagnosis by focusing on distribution quantiles and addressing limitations of existing methods.

Findings

The new $ ilde{R}_ ext{infty}$ effectively detects convergence issues in different quantiles.

The localized $ ilde{R}$ provides better insight into chain convergence behavior.

Theoretical properties support the use of the new indicators.

Abstract

Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named $\hat{R}$, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the $\hat{R}$ behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator $\hat{R}_\infty$, which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other $\hat{R}$ versions.