Title: Bounding the convergence rate of parallel tempering on multimodal
distributions

Abstract: Multimodal posterior distributions are commonly encountered
in Bayesian statistics.  However, one standard tool for sampling from
a posterior distribution, namely Metropolis-Hastings with local
proposals, can become stuck in local modes.  Parallel tempering is a
modification of Metropolis-Hastings that is often used in hopes of
circumventing bottlenecks between modes.

This approach has been shown to be successful for several symmetric
bimodal examples, including the mean field Ising model, and to be
unsuccessful on one trimodal example.  We obtain a general bound on
the convergence rate of parallel tempering that reflects its ability
to move between the modes of the distribution.  This bound leads to a
set of sufficient conditions for rapid mixing of parallel tempering on
multimodal distributions.  We show that parallel tempering on the
symmetric bimodal examples previously analyzed satisfies these
conditions, and illustrate which one of the conditions fails for the
trimodal example.