Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over Wasserstein distance. In this paper we prove that the infinite-dimensionality of the space of probabilities drastically deteriorates its sample complexity, which is slower than any polynomial rate in the sample size. We thus propose a new distance that preserves many desirable properties of the former while achieving a parametric rate of convergence. In particular, our distance 1) metrizes weak convergence; 2) can be estimated numerically through samples with low complexity; 3) can be bounded analytically from above and below. The main ingredient are integral probability metrics, which lead to the name hierarchical IPM.

翻译：随机概率是统计学和机器学习中许多非参数方法的核心组成部分。为了量化不同随机概率定律之间的比较，一些研究开始采用优雅的Wasserstein上Wasserstein距离。本文证明概率空间的无穷维特性会严重降低其样本复杂度，使其慢于样本量的任何多项式速率。为此，我们提出一种新距离，它在保持前者许多优良性质的同时，实现了参数收敛速度。特别地，本文提出的距离：1）可度量弱收敛；2）可通过低复杂度样本进行数值估计；3）可解析地给出上下界。其主要构成是积分概率度量，由此得名层级积分概率度量。