Conditional Diffusion Models Based Conditional Independence Testing

TL;DR

This paper introduces a new conditional independence testing method using conditional diffusion models to better approximate the conditional distribution, improving test accuracy especially in high-dimensional settings.

Contribution

It proposes using conditional diffusion models for more accurate approximation of $X|Z$ in CRT, outperforming GANs and handling complex, mixed-type data without distributional assumptions.

Findings

CDMs closely approximate the true conditional distribution.

The proposed test controls type I error effectively.

The method performs well in high-dimensional synthetic data experiments.

Abstract

Conditional independence (CI) testing is a fundamental task in modern statistics and machine learning. The conditional randomization test (CRT) was recently introduced to test whether two random variables, $X$ and $Y$, are conditionally independent given a potentially high-dimensional set of random variables, $Z$. The CRT operates exceptionally well under the assumption that the conditional distribution $X|Z$ is known. However, since this distribution is typically unknown in practice, accurately approximating it becomes crucial. In this paper, we propose using conditional diffusion models (CDMs) to learn the distribution of $X|Z$. Theoretically and empirically, it is shown that CDMs closely approximate the true conditional distribution. Furthermore, CDMs offer a more accurate approximation of $X|Z$ compared to GANs, potentially leading to a CRT that performs better than those based on GANs. To accommodate complex dependency structures, we utilize a computationally efficient classifier-based conditional mutual information (CMI) estimator as our test statistic. The proposed testing procedure performs effectively without requiring assumptions about specific distribution forms or feature dependencies, and is capable of handling mixed-type conditioning sets that include both continuous and discrete variables. Theoretical analysis shows that our proposed test achieves a valid control of the type I error. A series of experiments on synthetic data demonstrates that our new test effectively controls both type-I and type-II errors, even in high dimensional scenarios.