A Wasserstein Index of Dependence for Random Measures

Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between Lévy measures to solve a statistical problem. Dependent Bayesian nonparametric models provide flexible inference on distinct, yet related, groups of observations. Each component of a vector of random measures models a group of exchangeable observations, while their dependence regulates the borrowing of information across groups. We derive the first statistical index of dependence in [0,1] for (completely) random measures that accounts for their whole infinite-dimensional distribution, which is assumed to be equal across different groups. This is accomplished by using the geometric properties of the Wasserstein distance to solve a max–min problem at the level of the underlying Lévy measures. The Wasserstein index of dependence sheds light on the models’ deep structure and has desirable properties: (i) it is 0 if and only if the random measures are independent; (ii) it is 1 if and only if the random measures are completely dependent; (iii) it simultaneously quantifies the dependence of d≥2 random measures, avoiding the need for pairwise comparisons; (iv) it can be evaluated numerically. Moreover, the index allows for informed prior specifications and fair model comparisons for Bayesian nonparametric models. Supplementary materials for this article are available online.

最优传输（Optimal Transport）与瓦瑟斯坦距离（Wasserstein Distance）作为比较与关联随机结构的工具，正于诸多科学领域蓬勃发展。本文首次开创性地利用莱维测度（Lévy Measure）间的最优传输距离解决一项统计问题。相依贝叶斯非参数模型（Bayesian Nonparametric Model）可针对不同但相关的观测组实现灵活的统计推断：随机测度（Random Measure）向量的每个分量对应一组可交换观测，而测度间的相依性用于调控组间的信息共享。我们首次推导了适用于（完全）随机测度的[0,1]区间内相依性统计指标，该指标可刻画随机测度的完整无限维分布，且假设不同组的该分布保持一致。该指标的构建借助瓦瑟斯坦距离的几何特性，通过求解底层莱维测度层面的极大极小问题得以实现。该瓦瑟斯坦相依性指标可揭示模型的深层结构，且具备诸多优良性质：（i）当且仅当随机测度相互独立时，指标值为0；（ii）当且仅当随机测度完全相依时，指标值为1；（iii）可同时量化d≥2个随机测度间的相依性，无需进行两两比较；（iv）可通过数值方法计算得到。此外，该指标可为贝叶斯非参数模型提供合理的先验设定依据，并支持公平的模型比较。本文的补充材料可在线获取。