Testing Conditional Mean Independence Using Generative Neural Networks

TL;DR

This work develops a fully nonparametric conditional mean independence test for multivariate responses by introducing a population measure $\Gamma^*$ that characterizes $H_0: \mathbb{E}[Y|X,Z] = \mathbb{E}[Y|Z]$. The authors construct a sample statistic $\widehat{T}_n$ using cross-fitting, RKHS-based embeddings, and a conditional generator (GMMN) to obtain samples from $P_{X|Z}$, with a wild bootstrap to calibrate the null distribution. The method achieves double robustness to slow nonparametric estimation errors and demonstrates strong finite-sample performance in high-dimensional settings, including imaging data, while maintaining power against local alternatives at the parametric rate. The approach is validated through simulations and two imaging tasks (facial expression recognition and age estimation), illustrating practical applicability to high-dimensional covariates and multivariate outcomes. Overall, the paper offers a principled, scalable CMI testing framework that provides reliable size control and competitive power without restrictive parametric assumptions.

Abstract

Conditional mean independence (CMI) testing is crucial for statistical tasks including model determination and variable importance evaluation. In this work, we introduce a novel population CMI measure and a bootstrap-based testing procedure that utilizes deep generative neural networks to estimate the conditional mean functions involved in the population measure. The test statistic is thoughtfully constructed to ensure that even slowly decaying nonparametric estimation errors do not affect the asymptotic accuracy of the test. Our approach demonstrates strong empirical performance in scenarios with high-dimensional covariates and response variable, can handle multivariate responses, and maintains nontrivial power against local alternatives outside an $n^{-1/2}$ neighborhood of the null hypothesis. We also use numerical simulations and real-world imaging data applications to highlight the efficacy and versatility of our testing procedure.

Figures (4)

• Figure 1: Box plot of the p-values (left y-axis) and the test accuracies (red line, right y-axis) for different HRs. The blue dashed line represents the baseline accuracy. The red dashed line represents the 5% nominal level. The test accuracy for the face-covered case (face acc: 37.60) is shown at the bottom right corner.
• Figure 2: Original facial images in FER2013 (first column) and the covered images with HRs: TL, nose, right eye, mouth, left eye, eyes, face (Columns 2-8).
• Figure 3: Box plot of the p-values (left y-axis) and the test MAE (red line, right y-axis) for different HRs. The blue dashed line represents the baseline MAE. The red dashed line represents the 5% nominal level. The test MAE for the face-covered case (face MAE: 8.42) is shown at the bottom right corner.
• Figure 4: Original facial images in UTKFace (first column) and the covered images with HRs: TL, nose, mouth, eyes, face (Columns 2-6).

Theorems & Definitions (8)

• Proposition 1
• Remark 2
• Proposition 3
• Remark 5
• Theorem 6
• Remark 8
• Theorem 9
• Theorem 10