Conditions for Torpid Mixing of Parallel and Simulated Tempering on Multimodal Distributions

We obtain upper bounds on the convergence rates of Markov chains constructed by parallel and simulated tempering. These bounds are used to provide a set of sufficient conditions for torpid mixing of both techniques. We apply these conditions to show torpid mixing of parallel and simulated tempering for three examples: a normal mixture model with unequal covariances in RM and the mean-field Potts model with q ≥ 3, regardless of the number and choice of temperatures, and the meanfield Ising model when an insufficient set of temperatures is chosen. The latter result contrasts with the rapid mixing of parallel and simulated tempering on the meanfield Ising model with a linearly increasing set of temperatures as shown previously.