A Convergence Diagnostic for Bayesian Clustering

TL;DR

This paper introduces a new single-chain convergence diagnostic for Bayesian clustering MCMC, using a Hotelling-type statistic and regenerative sampling to reliably assess convergence in discrete spaces.

Contribution

It develops a novel convergence diagnostic tailored for discrete-space MCMC in Bayesian clustering, addressing limitations of existing multi-chain methods.

Findings

Effective in detecting non-convergent chains in Bayesian clustering

Utilizes unnormalized posterior to improve diagnostic accuracy

Demonstrated on genetic clustering of Arabidopsis thaliana

Abstract

In many applications of Bayesian clustering, posterior sampling on the discrete state space of cluster allocations is achieved via Markov chain Monte Carlo (MCMC) techniques. As it is typically challenging to design transition kernels to explore this state space efficiently, MCMC convergence diagnostics for clustering applications is especially important. For general MCMC problems, state-of-the-art convergence diagnostics involve comparisons across multiple chains. However, single-chain alternatives can be appealing for computationally intensive and slowly-mixing MCMC, as is typically the case for Bayesian clustering. Thus, we propose here a single-chain convergence diagnostic specifically tailored to discrete-space MCMC. Namely, we consider a Hotelling-type statistic on the highest probability states, and use regenerative sampling theory to derive its equilibrium distribution. By leveraging information from the unnormalized posterior, our diagnostic protects against seemingly convergent chains in which the relative frequency of visited states is incorrect. The methodology is illustrated with a Bayesian clustering analysis of genetic mutants of the flowering plant Arabidopsis thaliana.