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 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：随机概率是统计学与机器学习中许多非参数方法的核心组成部分。为量化不同随机概率律之间的比较，已有若干研究开始采用优雅的“Wasserstein距离之上的Wasserstein距离”。本文证明，概率空间的无限维特性会严重恶化其样本复杂度，其收敛速率慢于样本量的任意多项式阶。我们提出一种新的距离度量，在保持前者诸多优良性质的同时，能够达到参数级的收敛速率。具体而言，所提距离具有以下特性：1) 能够度量化弱收敛；2) 可通过样本以较低复杂度进行数值估计；3) 可从解析上给出其上下界。该方法的核心要素是积分概率度量，故命名为分层IPM。