We give conditions under which a Markov chain constructed via parallel or simulated tempering is guaranteed to be rapidly mixing, which are applicable to a wide range of multimodal distributions arising in Bayesian statistical inference and statistical mechanics. We provide lower bounds on the spectral gaps of parallel and simulated tempering. These bounds imply a single set of sufficient conditions for rapid mixing of both techniques. A direct consequence of our results is rapid mixing of parallel and simulated tempering for several normal mixture models, and for the mean-field Ising model.
"Conditions for rapid mixing of parallel and simulated tempering on multimodal distributions." Ann. Appl. Probab. 19 (2) 617 - 640, April 2009. https://doi.org/10.1214/08-AAP555