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.

A Wasserstein Index of Dependence for Random Measures / Catalano, Marta; Lavenant, Hugo; Lijoi, Antonio; Pruenster, Igor. - In: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION. - ISSN 0162-1459. - 119:547(2024), pp. 2396-2406. [10.1080/01621459.2023.2258596]

A Wasserstein Index of Dependence for Random Measures

Catalano, Marta;
2024

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 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.
2024
Bayesian nonparametrics; Index of dependence; Lévy measure; Random measure; Wasserstein distance
A Wasserstein Index of Dependence for Random Measures / Catalano, Marta; Lavenant, Hugo; Lijoi, Antonio; Pruenster, Igor. - In: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION. - ISSN 0162-1459. - 119:547(2024), pp. 2396-2406. [10.1080/01621459.2023.2258596]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11385/245038
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