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 ${\mathcal{L}}_{X|Z}$, ${\mathcal{L}}_{Y|Z}$, and ${\mathcal{L}}_{(X,Y)|Z}$, 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.

arXiv comment31 pages. v2 contains major revision to Section 3, plus assorted expository improvements

References in corpus (2)

• Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces
• Faster Wasserstein Distance Estimation with the Sinkhorn Divergence