Rank-normalization, folding, and localization: An improved $\widehat{R}$ for assessing convergence of MCMC (1903.08008v5)

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.

• The paper introduces a rank-normalized diagnostic that robustly assesses MCMC convergence in heavy-tailed settings.
• The paper incorporates quantile-based local efficiency measures to provide detailed Monte Carlo error estimates for quantiles.
• The paper employs folded statistics and rank plots to enhance visualization and detect convergence issues in chain variances.

An Improved Diagnostic for MCMC Convergence: Rank-Normalization and Efficiency Measures

This paper presents a significant enhancement in the monitoring of convergence for Markov chain Monte Carlo (MCMC) methods, a crucial computational tool in Bayesian statistical analysis. Traditional approaches to convergence diagnostics, particularly the use of the potential scale reduction factor (commonly referred to as R) from Gelman and Rubin (1992), possess notable deficiencies. In particular, these traditional diagnostics are inadequate in handling cases where the chain exhibits heavy tailed distributions or when there is significant variance between chains. The authors propose a new diagnostic method based on rank-normalization and introduce localized efficiency measures to address these limitations.

Key Contributions

1. Rank-Normalized Diagnostic: The paper introduces a rank-based diagnostic measure designed to assess the convergence of MCMC simulations more reliably. By using rank-normalized values instead of the raw chain outputs, the proposed method becomes robust to heavy-tailed distributions. This refinement allows the diagnostic to remain effective even when traditional variance-based methods fail.
2. Local Efficiency Measures: In addition to the rank-based diagnostic, the paper proposes quantile-based local efficiency measures which offer a more detailed understanding of a chain’s behavior. These measures provide practical estimates of Monte Carlo error for quantiles, a notable improvement over traditional methods focused primarily on posterior means.
3. Folded Statistics: The authors also introduce folded statistics that assess the convergence in the tails of the distribution. This is an innovative approach that calculates the absolute deviations from the median and uses these in convergence diagnostics to identify discrepancies in variance exploration between chains.
4. Rank Plots: The paper suggests replacing traditional trace plots with rank plots to better visualize convergence. Rank plots mitigate the interpretation challenges of traditional plots by providing a clear visual indication of mixing uniformity across chains.

Implications and Future Directions

The methodological improvements proposed in this paper have significant implications for computational Bayesian analysis where R\ has been an industry standard for convergence diagnostics. By addressing the deficiencies in traditional methods, the new diagnostics provide users with more reliable tools for assessing MCMC convergence, particularly in complex models with potential multimodality or heavy-tailed distributions.

From a practical perspective, the enhanced diagnostics could lead to more accurate statistical inference by preventing false convergence conclusions, thereby reducing the risk of erroneous results in real-world applications. For theoretical advancements, these tools open pathways for more robust convergence theories that do not rely on assumptions of finite variance or mean.

Moving forward, the paper hints at interesting research avenues such as extensive parallelization in MCMC methods, where convergence diagnostics might need further adaptation and refinement to remain effective with massive numbers of chains.

Conclusion

Overall, this paper makes a noteworthy contribution by proposing a rank-normalization based diagnostic and localized measures of efficiency that enhance the reliability of MCMC convergence assessments. While the traditional R\ diagnostic continues to be widely used, the insights and tools offered in this paper represent a step forward in the evolution of MCMC convergence diagnostics, attaching higher accuracy and reliability to Bayesian computational analyses. The authors’ work lays a foundation for continued improvement and adaptation of these methods in the face of growing computational complexity and high-dimensional data landscapes in Bayesian inference.