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Computational couplings of Markov chains provide a practical route tounbiased Monte Carlo estimation that can utilize parallel computation. However,these approaches depend crucially on chains meeting after a small number oftransitions. For models that assign data into groups, e.g. mixture models, theobvious approaches to couple Gibbs samplers fail to meet quickly. This failureowes to the so-called "label-switching" problem; semantically equivalentrelabelings of the groups contribute well-separated posterior modes that impedefast mixing and cause large meeting times. We here demonstrate how to avoidlabel switching by considering chains as exploring the space of partitionsrather than labelings. Using a metric on this space, we employ an optimaltransport coupling of the Gibbs conditionals. This coupling outperformsalternative couplings that rely on labelings and, on a real dataset, providesestimates more precise than usual ergodic averages in the limited time regime.Code is available at github.com/tinnguyen96/coupling-Gibbs-partition.

Optimal transport couplings of Gibbs samplers on partitions for unbiased estimation