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 normalizing constants of probability models. The method was originally proposed for cases where the probability models have overlapping support. Voter proposed the idea of shifting physical systems before applying the acceptance ratio method to calculate free-energy differences between systems that are highly separated in a configuration 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 bridge sampling 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 information item factor model and a nonlinear mixed model are provided to demonstrate the potentially substantial gains in Monte Carlo efficiency by going beyond centering and by using efficient 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.