Wasserstein Conditional Independence Testing

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 LX|Z, LY|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.