Wasserstein Conditional Independence Testing

Abstract

We introduce a test for the conditional independence of random variables X and Y given a random variable Z, specifically by sampling from the joint distribution (X, Y, Z), binning the support of the distribution of Z, and conducting multiple p-Wasserstein two-sample tests. Under a p-Wasserstein Lipschitz assumption on the conditional distributions L X|Z , L Y |Z , and L (X,Y )|Z , we show that it is possible to control the Type I and Type II error of this test, and give examples of explicit finite-sample error bounds in the case where the distribution of Z has compact support.