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Testing Conditional Mean Independence Using Generative Neural Networks

ICML 2025 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Likelihood-free and graphical model estimation · 0 citations (OpenAlex)

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Proposes a conditional mean independence test using a population measure in RKHS form, cross-fitted generative moment matching networks for the conditional distribution of X given Z and DNNs for E[Y|Z], and a wild bootstrap, with size control and power against n^{-1/2} local alternatives.

Problem

Testing conditional mean independence of Y and X given Z (multivariate Y, high-dimensional covariates) with accurate size control and power against local alternatives, where nonparametric nuisance estimation errors (from DNNs or generative models) threaten the asymptotic null distribution.

Approach

A multiplicative population CMI measure built from centered responses and kernel-based conditional-mean-embedding differences in X, so that nuisance estimation errors enter only as products (double robustness), estimated with a GMMN conditional generator and DNN regression under cross-fitting, with a Rademacher wild bootstrap for the critical values.

Key results

• In Example A1 under H1 sparse at n=400, the proposed test achieves 100% size-adjusted power versus 27.0% for VIM and 81.8% for PCM, while keeping empirical size near 5-9% under H0.
• In Example A2 under sparse alternatives at n=800, the proposed test reaches 100% power, comparable to DSP-based tests, while DSP_M and DSP achieve best power at n=200-400.
• On FER2013, p-values from the proposed test lie above the 5% level only when nose or top-left corner is covered, consistent with VGG accuracies near baseline in those cases.

Measured effects reported in the paper

Surprising findings

• The pMIT_M test with DNN estimation has large size distortions on UTKFace, with all p-values below the nominal level even when the top-left-covered accuracy is near baseline.
• The DSP_M statistic with cross-entropy loss yields p-values below 5% for regions that should be null (top-left corner or nose), suggesting inflated type-I error.

Limitations

• Power vanishes for alternatives within the n^{-1/2} neighborhood of the null (alpha > 1/2), so the test has trivial power for very local alternatives.
• Theoretical size guarantees depend on nuisance error rates satisfying alpha1+alpha2 > 1/2, and total variation metrics may be unsuitable for nearly singular distributions such as images.
• Real-data results are based on a limited set of hypothesized regions and subsamples of size n=2000, and conditioning on a fixed-architecture classifier as a proxy for the true null.

Experimental setup

Datasets: FER2013 UTKFace simulated linear and nonlinear regression data

Benchmarks: simulated Examples A1 and A2 FER2013 facial expression application UTKFace age application

Metrics: empirical size size-adjusted power p-values test accuracy MAE

Baselines: VIM PCM PCM_M DSP DSP_M pMIT pMIT_M pMIT_e pMIT_eM

Builds on: maximum mean discrepancy conditional independence test (MMDCI) power enhanced conditional independence test noise-outsourcing lemma generative moment matching networks wild bootstrap

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