Dynamic Tempered Transitions for Exploring Multimodal Posterior Distributions

Multimodal, high-dimension posterior distributions are well known to cause mixing problems for standard Markov chain Monte Carlo (MCMC) procedures; unfortunately such functional forms readily occur in empirical political science. This is a particularly important problem in applied Bayesian work because inferences are made from finite intervals of the Markov chain path. To address this issue, we develop and apply a new MCMC algorithm based on tempered transitions of simulated annealing, adding a dynamic element that allows the chain to self-tune its annealing schedule in response to current posterior features. This important feature prevents the Markov chain from getting trapped in minor modal areas for long periods of time. The algorithm is applied to a probabilistic spatial model of voting in which the objective function of interest is the candidate's expected return. We first show that such models can lead to complex target forms and then demonstrate that the dynamic algorithm easily handles even large problems of this kind.