Maximal leakage quantifies the leakage of information from data $X \in \mathcal{X}$ due to an observation $Y$. While fundamental properties of maximal leakage, such as data processing, sub-additivity, and its connection to mutual information, are well-established, its behavior over Bayesian networks is not well-understood and existing bounds are primarily limited to binary $\mathcal{X}$. In this paper, we investigate the behavior of maximal leakage over Bayesian networks with finite alphabets. Our bounds on maximal leakage are established by utilizing coupling-based characterizations which exist for channels satisfying certain conditions. Furthermore, we provide more general conditions under which such coupling characterizations hold for $|\mathcal{X}| = 4$. In the course of our analysis, we also present a new simultaneous coupling result on maximal leakage exponents. Finally, we illustrate the effectiveness of the proposed bounds with some examples.

翻译：最大泄漏量化了由于观测变量 $Y$ 导致数据 $X \in \mathcal{X}$ 的信息泄漏程度。尽管最大泄漏的基本性质（如数据处理、次可加性及其与互信息的联系）已得到充分确立，但其在贝叶斯网络上的行为尚未被充分理解，且现有界主要局限于二元 $\mathcal{X}$。本文研究了有限字母表下贝叶斯网络上最大泄漏的行为。我们通过利用满足特定条件的信道所存在的基于耦合的表征，建立了最大泄漏的界。此外，针对 $|\mathcal{X}| = 4$ 的情况，我们给出了此类耦合表征成立的更一般条件。在分析过程中，我们还提出了关于最大泄漏指数的新同步耦合结果。最后，通过若干示例说明了所提界的有效性。