On simulated tempering for multimodal targets

Oregon Convention Center 

Simulated tempering Markov chain Monte Carlo method has long been an important tool for exploring multimodal distributions that are difficult to sample from. The method creates 'tempered' distributions that help to move the Markov chain between modes of the target density. While one can ignore the samples not coming from the target distribution, an importance sampling estimator that uses all the samples generated is usually more numerically stable. The current practice requires repeated pilot runs to estimate normalizing constants of these 'tempered' distributions, yet often overlooks the inherent errors of those estimates, which leads to under-reporting of the standard errors. We provide an effective method for implementing simulated tempering efficiently using the available computing resources and derive formulas for asymptotically valid standard errors of the estimators. The proposed methods are illustrated using several examples.