MEASURING DEPENDENCE IN THE WASSERSTEIN DISTANCE FOR BAYESIAN NONPARAMETRIC MODELS

• 成果类型: Article
• 署名作者: Catalano, Marta; Lijoi, Antonio; Prunster, Igor
• 署名单位: University of Warwick; Bocconi University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2065
• 发表日期: 2021
• 页码: 2916-2947
• 关键词: priors

摘要

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 Frechet 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 Levy intensities are derived relying on compound Poisson approximations. These are then specialized to popular models in the Bayesian literature leading to interesting insights.