Erwin Schrödinger posed—and to a large extent solved—in 1931/32 the problem of finding the most likely random evolution between two continuous probability distributions. This article considers this problem in the case when only samples of the two distributions are available. A novel iterative procedure is proposed, inspired by Fortet‐IPF‐Sinkhorn type algorithms. Since only samples of the marginals are available, the new approach features constrained maximum likelihood estimation in place of the nonlinear boundary couplings, and importance sampling to propagate the functions ϕ and solving the Schrödinger system. This method mitigates the curse of dimensionality, compared to the introduction of grids, which in high dimensions lead to numerically unfeasible methods. The methodology is illustrated in two applications: entropic interpolation of two‐dimensional Gaussian mixtures, and the estimation of integrals through a variation of importance sampling. © 2020 Wiley Periodicals LLC.

中文翻译：

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数据驱动的薛定ding桥

欧文·薛定er（ErwinSchrödinger）在1931/32提出并在很大程度上解决了寻找两个连续概率分布之间最可能的随机演化的问题。当只有两个分布的样本可用时，本文将考虑此问题。在Fortet-IPF-Sinkhorn类型算法的启发下，提出了一种新颖的迭代过程。由于只有边际样本可用，因此新方法具有约束最大似然估计的功能，代替了非线性边界耦合，并且具有重要抽样以传播函数ϕ和解决薛定ding系统。与网格的引入相比，此方法减轻了维数的诅咒，而网格的引入导致了数字上不可行的方法。该方法在两个应用中得到了说明：二维高斯混合的熵内插，以及通过重要度采样的变化来估计积分。©2020 Wiley Periodicals LLC。