Bounds on Maximal Leakage over Bayesian Networks

TL;DR

This paper explores the behavior of maximal leakage in Bayesian networks with finite alphabets, providing new bounds and coupling characterizations that extend previous binary-focused results.

Contribution

It introduces bounds on maximal leakage over Bayesian networks with finite alphabets and generalizes coupling characterizations beyond binary cases.

Findings

Established bounds on maximal leakage using coupling methods.

Extended coupling characterizations to alphabets of size 4.

Presented a new coupling result on maximal leakage exponents.

Abstract

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.