  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.
