The MCMC convergence graph: a diagnostic for multimodal posteriors

2026-09-29

AI Summary

This study addresses the limitation of standard MCMC diagnostics, which frequently misinterpret multimodal posteriors as sampling failures. To overcome this, it proposes a graph-theoretic convergence diagnostic framework that constructs a graph structure by computing between-chain R-hat values, thereby identifying groups of Markov chains exploring the same posterior mode. The method is implemented using Stan and the R package mcmcConvergenceGraph. As the first approach to leverage graph structures for distinguishing genuine multimodality from poor mixing, it prevents the penalizing misdiagnosis of multimodal posteriors. By providing both graphical and numerical summaries, the framework offers comprehensive diagnostic insights. Its effectiveness in accurately diagnosing multimodal behavior is validated through applications in regression and pharmacokinetic models.

Abstract

Multimodality is common in many scientific and engineering applications. However, standard Markov chain Monte Carlo (MCMC) convergence diagnostics often penalise multimodality in practice by conflating overall poor mixing with groups of chains that have mixed well within distinct modes. To diagnose multimodality rather than flag it as a sampling failure, we introduce the MCMC convergence graph: a graph-based diagnostic that computes pairwise R-hat values between chains and summarises them in a graph, revealing sets of chains that explore the same posterior mode. We implement the method in the probabilistic programming framework Stan and demonstrate its use on worked examples ranging from regression models with inherent multimodality in the data, to a pharmacokinetic model for which multimodality reflects non-identifiability. The graph-based diagnostic, which we make freely available in the R package mcmcConvergenceGraph and which accepts MCMC output from any sampler, provides both graphical and numerical summaries of MCMC behaviour for diagnosing multimodality.

Problem

• MCMC convergence diagnostics
• multimodal posteriors
• Markov chain Monte Carlo
• R-hat

Innovation

• MCMC convergence graph
• multimodal posteriors
• pairwise R-hat
• graph-based diagnostic
• Markov chain Monte Carlo