Warp Bridge Sampling

Bridge sampling, a general formulation of the acceptance ratio method in physics for computing free-energy difference, is an effective Monte Carlo method for computing normalizingconstantsof probabilitymodels.The methodwas originallyproposedfor cases where the probabilitymodels have overlappingsupport. Voter proposed the idea of shifting physical systems before applying the acceptance ratio method to calculate free-energy differencesbetween systems that are highlyseparatedin a conŽ guration space.The purpose of this article is to push Voter’s idea further by applying more general transformations, including stochastic transformations resulting from mixing over transformation groups, to the underlying variables before performing bridge sampling.We term such methods warp bridgesampling to highlight the fact that in addition to location shifting (i.e., centering)one can further reduce the difference/distance between two densities by warping their shapes without changing the normalizing constants. Real data-based empirical studies using the full informationitem factormodeland a nonlinearmixedmodel are providedto demonstrate the potentially substantial gains in Monte Carlo efŽ ciency by going beyond centering and by using efŽ cient bridge sampling estimators. Our general method is also applicable to a couple of recent proposals for computing marginal likelihoods and Bayes factors because these methods turn out to be covered by the general bridge sampling framework.