Title: Differentially Private Joint Independence Test

Abstract: Detecting joint dependence among multiple random vectors is a critical task in numerous statistical applications, particularly when dealing with sensitive or confidential data. This study addresses the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) within the framework of differential privacy. Since the asymptotic distribution of the empirical dHSIC estimate involves complex Gaussian chaos, non-private testing procedures typically rely on permutation and bootstrap techniques. To address joint dependence under privacy constraints, we introduce a testing procedure based on dHSIC that utilizes a differentially private permutation approach. Our analysis demonstrates that this method ensures privacy protection, maintains correct test levels, and achieves pointwise consistency, in contrast to bootstrap-based alternatives which exhibit inconsistent power. Furthermore, we examine the uniform power of our proposed test under both $dHSIC$ and $L_2$ metrics, proving that it reaches minimax optimal power across various privacy regimes. As a secondary finding, we establish that the non-private permutation dHSIC test introduced by Pfister et al. (2018) is a specific instance of our differentially private permutation test. Consequently, our results confirm the pointwise and uniform power of the Pfister et al. method, thereby resolving an open issue raised in their original work. Empirical evaluations through numerical simulations and real-world causal inference data analyses indicate that our proposed test performs effectively.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC