A Wasserstein index of dependence for random measures

TL;DR

This paper introduces a novel Wasserstein-based dependence index for random measures, enabling comprehensive, numerical quantification of dependence in Bayesian nonparametric models across multiple groups.

Contribution

It develops the first statistical dependence index for random measures using Wasserstein distances, capturing the entire infinite-dimensional structure and applicable to multiple measures simultaneously.

Findings

The index is 0 if and only if measures are independent.

The index is 1 if and only if measures are fully dependent.

It can be computed numerically for practical use.

Abstract

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 L\'{e}vy 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 L\'{e}vy 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 $d \ge 2$ 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.