A Wasserstein Index of Dependence for Random Measures

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 $d\ge 2$ 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. Supplementary materials for this article are available online.

Keywords:

• Bayesian nonparametrics
• Index of dependence
• Lévy measure
• Random measure
• Wasserstein distance

Supplementary Materials

The Supplementary Material contains our proof techniques and the underlying optimal transport problem, which we believe are of interest beyond the present setup with natural applications to the theory of partial differential equations and of Lévy processes.

Acknowledgments

The authors would like to thank the Editor, an Associate Editor, and four Referees for constructive comments which led to a substantial improvement and expansion of the article. In particular, one of the Referees has inspired the analysis carried out in Section 5.3. Moreover, the authors are grateful to Giuseppe Savaré for helpful discussions and to Riccardo Corradin for valuable insights on the simulations in Section 6. Part of this work was carried out while Marta Catalano was affiliated with the Department of Statistics of the University of Warwick.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

M. Catalano was partially supported by the Heilbronn Institute for Mathematical Research, A. Lijoi and I. Prünster by MIUR, PRIN Project 2022CLTYP4.