A Wasserstein Index of Dependence for Random Measures

• 成果类型: Article
• 署名作者: Catalano, Marta; Lavenant, Hugo; Lijoi, Antonio; Prunster, Igor
• 署名单位: Luiss Guido Carli University; Bocconi University; Bocconi University; Bocconi University; Bocconi University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2023.2258596
• 发表日期: 2024
• 页码: 2396-2406
• 关键词: inference

摘要

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 Levy 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 Levy 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.