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.

摘要

桥接采样是一种有效的蒙特卡罗 (MC) 方法，用于估计两个概率密度的归一化常数之比，是统计学、物理、化学和其他领域的常规计算问题。桥采样估计器的 MC 误差由两个密度之间的重叠量决定。在单峰密度的情况下，Warp-I、II 和 III 变换对于增加初始重叠是有效的，但对于多峰密度则效果较差。本文介绍了 Warp-U 变换，旨在将多峰密度转换为单峰密度（因此是“U”）而不改变它们的归一化常数。Warp-U 变换的构造从正态（或其他方便的）混合分布开始${\phi }_{\text{混合}}$与目标密度p有合理的重叠，其归一化常数未知。映射的随机变换${\phi }_{\text{混合}}$回到它的生成分布$\mathcal{ñ}(0,1)$然后应用于p产生它的 Warp-U 版本，我们表示$\stackrel{～}{p}$. 通常，$\stackrel{～}{p}$是单峰的，并且与$\phi$. 此外，我们证明了$\stackrel{～}{p}$和$\mathcal{ñ}(0,1)$保证不小于p和${\phi }_{\text{混合}}$，就任何f -散度而言。我们提出了一种计算有效的方法来找到合适的${\phi }_{\text{混合}}$，以及一种简单但有效的方法来消除因估计归一化常数和拟合而产生的偏差${\phi }_{\text{混合}}$具有相同的数据。我们使用 10 维和 50 维高度不规则的多模态密度来说明我们的发现，并演示如何使用 Warp-U 采样来改进广义 Wang-Landau 算法的最终估计步骤，这是一种强大的采样和估计方法。本文的补充材料可在线获取。