Sufficient Conditions for Torpid Mixing of Parallel and Simulated Tempering

We obtain upper bounds on the spectral gap of Markov chains constructed by parallel and simulated tempering, and provide a set of sufficient conditions for torpid mixing of both techniques. Combined with the results of [22], these results yield a two-sided bound on the spectral gap of these algorithms. We identify a persistence property of the target distribution, and show that it can lead unexpectedly to slow mixing that commonly used convergence diagnostics will fail to detect. For a multimodal distribution, the persistence is a measure of how "spiky", or tall and narrow, one peak is relative to the other peaks of the distribution. We show that this persistence phenomenon can be used to explain the torpid mixing of parallel and simulated tempering on the ferromagnetic mean-field Potts model shown previously. We also illustrate how it causes torpid mixing of tempering on a mixture of normal distributions with unequal covariances in R-M, a previously unknown result with relevance to statistical inference problems. More generally, any-time a multimodal distribution includes both very narrow and very wide peaks of comparable probability mass, parallel and simulated tempering are shown to mix slowly.