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Wasserstein Conditional Independence Testing

Andrew Warren

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.

Wasserstein Conditional Independence Testing

Abstract

We introduce a test for the conditional independence of random variables and given a random variable , specifically by sampling from the joint distribution , binning the support of the distribution of , and conducting multiple -Wasserstein two-sample tests. Under a -Wasserstein Lipschitz assumption on the conditional distributions , , and , 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 has compact support.

Paper Structure

This paper contains 7 sections, 12 theorems, 140 equations.

Table of Contents

  1. Two-sample independence testing with the Wasserstein distance
  2. Conditional independence testing via binning
  3. Plug-in estimation of the Lipschitz constant
  4. Discussion
  5. Proof of Lemma \ref{['lem:disintegration']}
  6. Additional Results
  7. Concentration of $W_{p}^{p}(\mu_{n},\mu)$

Key Result

Proposition 2

Let $(X,d)$ be a Polish metric space, and let $\mu\in\mathcal{P}_{p}(X)$ be a measure supported on a set of diameter at most $D$. Let $\mu_{n}$ denote an empirical measure for $\mu$. Then,

Theorems & Definitions (35)

  • Definition 1
  • Remark
  • Proposition 2: weed2019sharp
  • Remark
  • Theorem 3: dereich2013constructive
  • Remark
  • Remark
  • Example 4
  • Example 5: Additive noise model; $Z$ causes $X$ and $Y$ but $X\perp\!\!\!\perp Y\mid Z$
  • Remark
  • ...and 25 more