# Rank-normalization, folding, and localization: An improved $\widehat{R}$ for assessing convergence of MCMC

## TL;DR

This paper identifies flaws in the traditional $\u0304R$ convergence diagnostic for MCMC, proposes a rank-based alternative, and introduces quantile-based measures for more reliable convergence assessment.

## Contribution

It introduces a robust rank-normalization diagnostic and local efficiency measures to improve convergence detection in MCMC, addressing issues with heavy tails and variance heterogeneity.

## Key findings

- The new diagnostic correctly identifies convergence failures missed by traditional $\u0304R$.
- Rank plots outperform trace plots for convergence assessment.
- Quantile-based measures provide reliable Monte Carlo error estimates.

## Abstract

Markov chain Monte Carlo is a key computational tool in Bayesian statistics, but it can be challenging to monitor the convergence of an iterative stochastic algorithm. In this paper we show that the convergence diagnostic $\widehat{R}$ of Gelman and Rubin (1992) has serious flaws. Traditional $\widehat{R}$ will fail to correctly diagnose convergence failures when the chain has a heavy tail or when the variance varies across the chains. In this paper we propose an alternative rank-based diagnostic that fixes these problems. We also introduce a collection of quantile-based local efficiency measures, along with a practical approach for computing Monte Carlo error estimates for quantiles. We suggest that common trace plots should be replaced with rank plots from multiple chains. Finally, we give recommendations for how these methods should be used in practice.