Conditional Diffusion Models Based Conditional Independence Testing
This work addresses a fundamental challenge in statistics and machine learning for researchers and practitioners, offering an incremental improvement over existing methods like GANs for conditional independence testing.
The paper tackles the problem of conditional independence testing when the conditional distribution is unknown by proposing conditional diffusion models to approximate it, showing theoretically and empirically that they closely match the true distribution and outperform GAN-based methods, with experiments on synthetic data demonstrating effective control of type-I and type-II errors in high-dimensional settings.
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.