2015/10/20 by Christophe Ley, Ley, Christophe, Gesine Reinert +3
Computer Science · Engineering · Mathematics · #Adversarial Robustness in Machine Learning #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1510.05826
openalex publication_date 2015/10/20 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this paper we propose tight upper and lower bounds for the Wasserstein\ndistance between any two univariate continuous distributions with\nprobability densities p1 and p2 having nested supports. These explicit\nbounds are expressed in terms of the derivative of the likelihood ratio\np1/p2 as well as the Stein kernel \τ1 of p1. The method of proof\nrelies on a new variant of Stein's method which manipulates Stein operators.\n We give several applications of these bounds. Our main application is in\nBayesian statistics : we derive explicit data-driven bounds on the Wasserstein\ndistance between the posterior distribution based on a given prior and the\nno-prior posterior based uniquely on the sampling distribution. This is the\nfirst finite sample result confirming the well-known fact that with\nwell-identified parameters and large sample sizes, reasonable choices of prior\ndistributions will have only minor effects on posterior inferences if the data\nare benign.\n