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October 2021 Measuring dependence in the Wasserstein distance for Bayesian nonparametric models
Marta Catalano, Antonio Lijoi, Igor Prünster
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Ann. Statist. 49(5): 2916-2947 (October 2021). DOI: 10.1214/21-AOS2065

Abstract

The proposal and study of dependent Bayesian nonparametric models has been one of the most active research lines in the last two decades, with random vectors of measures representing a natural and popular tool to define them. Nonetheless, a principled approach to understand and quantify the associated dependence structure is still missing. We devise a general, and not model-specific, framework to achieve this task for random measure based models, which consists in: (a) quantify dependence of a random vector of probabilities in terms of closeness to exchangeability, which corresponds to the maximally dependent coupling with the same marginal distributions, that is, the comonotonic vector; (b) recast the problem in terms of the underlying random measures (in the same Fréchet class) and quantify the closeness to comonotonicity; (c) define a distance based on the Wasserstein metric, which is ideally suited for spaces of measures, to measure the dependence in a principled way. Several results, which represent the very first in the area, are obtained. In particular, useful bounds in terms of the underlying Lévy intensities are derived relying on compound Poisson approximations. These are then specialized to popular models in the Bayesian literature leading to interesting insights.

Funding Statement

Antonio Lijoi and Igor Prünster are partially supported by MIUR, PRIN Project 2015SNS29B.

Acknowledgments

The authors are grateful to the Editor and two referees for insightful comments and remarks, which led to a substantial improvement of the manuscript. Special thanks are due to one of referees who suggested the relationship reported in Lemma 3. The paper was completed while Marta Catalano was a Ph.D. student at Bocconi University. The authors are also affiliated to the Bocconi Institute for Data Science and Analytics (BIDSA) and the Collegio Carlo Alberto.

Citation

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Marta Catalano. Antonio Lijoi. Igor Prünster. "Measuring dependence in the Wasserstein distance for Bayesian nonparametric models." Ann. Statist. 49 (5) 2916 - 2947, October 2021. https://doi.org/10.1214/21-AOS2065

Information

Received: 1 June 2020; Revised: 1 December 2020; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/21-AOS2065

Subjects:
Primary: 60G09 , 60G57 , 62C10

Keywords: Bayesian nonparametrics , completely random measures , completely random vectors , Compound Poisson , Dependence , Independent increments , Lévy copula , Wasserstein distance

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 5 • October 2021
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