Identification of joint dependence among several random vectors plays an important role in many statistical applications, where the data may contain sensitive or confidential information. In this paper, we consider the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) in the context of differential privacy. Given that the limiting distribution of the empirical estimate of dHSIC is a complicated Gaussian chaos, constructing tests in the non-private regime is typically based on permutation and bootstrap methods. To detect joint dependence under privacy constraints, we propose a dHSIC-based testing procedure employing a differentially private permutation methodology. We show that our method enjoys privacy guarantees, a valid level, and pointwise consistency, whereas the bootstrap counterpart suffers from inconsistent power. We further investigate the uniform power of the proposed test under the dHSIC and $L_2$ metrics, showing that the proposed test attains the minimax optimal power across different privacy regimes. As a byproduct, we show that the non-private permutation dHSIC test proposed in Pfister et al. (2018) is a special case of our differentially private permutation test, and our results also establish its pointwise and uniform power--thus resolving an open problem from that work. Both numerical simulations and real data analysis in causal inference suggest that our proposed test performs well empirically.

翻译：多个随机向量间的联合依赖性识别在众多统计应用中扮演重要角色，此时数据可能包含敏感或机密信息。本文在差分隐私框架下考虑$d$变量希尔伯特-施密特独立性准则（dHSIC）。鉴于dHSIC经验估计的极限分布是复杂的高斯混沌过程，非隐私场景下的检验通常基于置换法和自助法构建。为在隐私约束下检测联合依赖性，我们提出基于dHSIC的检验流程，采用差分隐私置换法。研究表明，该方法具有隐私保证、有效水平和逐点一致性，而自助法对应方法存在势函数不一致问题。我们进一步研究了所提检验在dHSIC和$L_2$度量下的均匀势，证明该方法在不同隐私机制下达到极小化最优势。作为副产品，我们证实Pfister等（2018）提出的非隐私置换dHSIC检验是本文差分隐私置换检验的特例，并同时建立其逐点势与均匀势——由此解决了该工作中的开放问题。因果推断中的数值模拟与真实数据分析均表明，所提检验具有良好实证表现。