Dynamic Tempered Transitions for Exploring Multimodal Posterior Distributions

George CasellaDepartment of Statistics, University of Florida, Griffin-Floyd Hall, P.O. Box 118545, Gainesville, FL 32611 e-mail: casella{at}stat.ufl.edu Multimodal, high-dimension posterior distributions are wellknown to cause mixing problems for standard Markov chain MonteCarlo (MCMC) procedures; unfortunately such functional formsreadily occur in empirical political science. This is a particularlyimportant problem in applied Bayesian work because inferencesare made from finite intervals of the Markov chain path. Toaddress this issue, we develop and apply a new MCMC algorithmbased on tempered transitions of simulated annealing, addinga dynamic element that allows the chain to self-tune its annealingschedule in response to current posterior features. This importantfeature prevents the Markov chain from getting trapped in minormodal areas for long periods of time. The algorithm is appliedto a probabilistic spatial model of voting in which the objectivefunction of interest is the candidate's expected return. Wefirst show that such models can lead to complex target formsand then demonstrate that the dynamic algorithm easily handleseven large problems of this kind.