  Markov chain Monte Carlo (MCMC) has transformed Bayesian model inference over
the past three decades: mainly because of this, Bayesian inference is now a
workhorse of applied scientists. Under general conditions, MCMC sampling
converges asymptotically to the posterior distribution, but this provides no
guarantees about its performance in finite time. The predominant method for
monitoring convergence is to run multiple chains and monitor individual chains'
characteristics and compare these to the population as a whole: if within-chain
and between-chain summaries are comparable, then this is taken to indicate that
the chains have converged to a common stationary distribution. Here, we
introduce a new method for diagnosing convergence based on how well a machine
learning classifier model can successfully discriminate the individual chains.
We call this convergence measure $R^*$. In contrast to the predominant
$\widehat{R}$, $R^*$ is a single statistic across all parameters that indicates
lack of mixing, although individual variables' importance for this metric can
also be determined. Additionally, $R^*$ is not based on any single
characteristic of the sampling distribution; instead it uses all the
information in the chain, including that given by the joint sampling
distribution, which is currently largely overlooked by existing approaches. We
recommend calculating $R^*$ using two different machine learning classifiers -
gradient-boosted regression trees and random forests - which each work well in
models of different dimensions. Because each of these methods outputs a
classification probability, as a byproduct, we obtain uncertainty in $R^*$. The
method is straightforward to implement and could be a complementary additional
check on MCMC convergence for applied analyses.
