Warp Bridge Sampling: The Next Generation

Abstract

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 ${\varphi }_{\text{mix}}$ that has reasonable overlap with the target density p, whose normalizing constant is unknown. The stochastic transformation that maps ${\varphi }_{\text{mix}}$ back to its generating distribution $\mathcal{N}(0,1)$ is then applied to p yielding its Warp-U version, which we denote $\tilde{p}$. Typically, $\tilde{p}$ is unimodal and has substantially increased overlap with $\varphi$. Furthermore, we prove that the overlap between $\tilde{p}$ and $\mathcal{N}(0,1)$ is guaranteed to be no less than the overlap between p and ${\varphi }_{\text{mix}}$, in terms of any f-divergence. We propose a computationally efficient method to find an appropriate ${\varphi }_{\text{mix}}$, and a simple but effective approach to remove the bias which results from estimating the normalizing constant and fitting ${\varphi }_{\text{mix}}$ 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.

Keywords:

• Bridge sampling
• Monte Carlo integration
• Normalizing constants
• Gaussian mixture
• Stochastic transformation
• Wang-Landau algorithm

Supplementary Materials

The online supplementary materials contain five appendices, R code implementing Warp-U bridge sampling for the 10 dimensional example presented in Sections 4 and 5, and the full version of the disclaimer given below.

This article is dedicated to the memory of Stephen G. Schilling (1958–2019), who made important contributions to the development of warp bridge sampling, as documented in Meng and Schilling (2002), as well as to his wife, Michele Desvignes-Schilling (1953–2014), a Star Trek fan, who inspired the adoption of the term “warp.”

Disclaimer

The views expressed herein are solely the views of the author(s) and are not necessarily the views of Two Sigma Investments, LP or any of its affiliates. They are not intended to provide, and should not be relied upon for, investment advice. Please see the full disclaimer on page 10 of the online supplementary materials.

Acknowledgments

The authors thank an anonymous associate editor and two anonymous referees for constructive comments and suggestions that improved the article. The authors also gratefully acknowledge helpful conversations with members of the Department of Statistics at Harvard University and at Texas A&M University, constructive comments from the audience of the 2016 MCQMC conference at Stanford University.

Additional information

Funding

The authors acknowledge partial financial support from NSF and JTF.