Optimal transport couplings of Gibbs samplers on partitions for unbiased estimation

Computational couplings of Markov chains provide a practical route to unbiased Monte Carlo estimation that can utilize parallel computation. However, these approaches depend crucially on chains meeting after a small number of transitions. For models that assign data into groups, e.g. mixture models, the obvious approaches to couple Gibbs samplers fail to meet quickly. This failure owes to the so-called "label-switching" problem; semantically equivalent relabelings of the groups contribute well-separated posterior modes that impede fast mixing and cause large meeting times. We here demonstrate how to avoid label switching by considering chains as exploring the space of partitions rather than labelings. Using a metric on this space, we employ an optimal transport coupling of the Gibbs conditionals. This coupling outperforms alternative couplings that rely on labelings and, on a real dataset, provides estimates more precise than usual ergodic averages in the limited time regime. Code is available at github.com/tinnguyen96/coupling-Gibbs-partition.