Sufficient Conditions for Torpid Mixing of Parallel and Simulated Tempering
Dawn B Woodard, Cornell University
Scott C Schmidler, Duke Universtity
Mark Huber, Duke University
Abstract
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 Woodard, Schmidler and Huber (2009), 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, anytime a multimodal distribution includes both very narrow
and very wide peaks of comparable probability mass, parallel and
simulated tempering are shown to mix slowly.
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