Measuring dependence in the Wasserstein distance for Bayesian nonparametric models
针对贝叶斯非参数模型中的随机测度向量，提出一种基于Wasserstein距离的通用框架来量化其依赖结构，通过逼近可交换性（共单调性）来度量依赖性，并利用复合泊松近似推导出基于Lévy强度的有用界限。
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.