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.

翻译：我们提出一种检验随机变量$X$与$Y$在给定随机变量$Z$下的条件独立性的方法，具体通过从联合分布$(X,Y,Z)$中采样，对$Z$分布的支撑集进行分箱，并执行多次$p$-Wasserstein双样本检验。在条件分布$\mathcal{L}_{X|Z}$、$\mathcal{L}_{Y|Z}$和$\mathcal{L}_{(X,Y)|Z}$满足$p$-Wasserstein Lipschitz假设的条件下，我们证明该检验能够控制第一类错误与第二类错误，并给出当$Z$分布具有紧支撑集时显式有限样本误差界的实例。