Warp Bridge Sampling: The Next Generation

Bridge sampling is an effective Monte Carlo (MC) method for estimating the ratio of normalizing constants of two probability densities, a routine computational problem in statistics, physics, chemistry, and other fields. The MC error of the bridge sampling estimator is determined by the amount of overlap between the two densities. In the case of unimodal densities, Warp-I, II, and III transformations are effective for increasing the initial overlap, but they are less so for multimodal densities. This article introduces Warp-U transformations that aim to transform multimodal densities into unimodal ones (hence “U”) without altering their normalizing constants. The construction of a Warp-U transformation starts with a normal (or other convenient) mixture distribution ϕmix that has reasonable overlap with the target density p, whose normalizing constant is unknown. The stochastic transformation that maps ϕmix back to its generating distribution N(0,1) is then applied to p yielding its Warp-U version, which we denote p˜ . Typically, p˜ is unimodal and has substantially increased overlap with ϕ . Furthermore, we prove that the overlap between p˜ and N(0,1) is guaranteed to be no less than the overlap between p and ϕmix , in terms of any f-divergence. We propose a computationally efficient method to find an appropriate ϕmix , and a simple but effective approach to remove the bias which results from estimating the normalizing constant and fitting ϕmix with the same data. We illustrate our findings using 10 and 50 dimensional highly irregular multimodal densities, and demonstrate how Warp-U sampling can be used to improve the final estimation step of the Generalized Wang–Landau algorithm, a powerful sampling and estimation approach. Supplementary materials for this article are available online.