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Bounds on Maximal Leakage over Bayesian Networks

Anuran Makur, Japneet Singh

TL;DR

The paper addresses bounding maximal leakage over Bayesian networks with finite alphabets using a coupling-based approach that leverages both network structure and per-channel properties.It introduces simultaneous coupling results for sets of marginal channels, derives upper bounds on the leakage exponent for composite channels, and presents a Doeblin-based simplification to bound leakage in Bayesian networks.A constructive coupling is provided for the case $|\mathcal{X}|=4$, extending existing results beyond the binary-input regime, and the framework is illustrated with simple network examples and potential extensions.The work opens avenues for generalizing to related leakage measures, relaxing key conditions, and deeper characterization of channels achieving bound equality.

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.

Bounds on Maximal Leakage over Bayesian Networks

TL;DR

The paper addresses bounding maximal leakage over Bayesian networks with finite alphabets using a coupling-based approach that leverages both network structure and per-channel properties.It introduces simultaneous coupling results for sets of marginal channels, derives upper bounds on the leakage exponent for composite channels, and presents a Doeblin-based simplification to bound leakage in Bayesian networks.A constructive coupling is provided for the case $|\mathcal{X}|=4$, extending existing results beyond the binary-input regime, and the framework is illustrated with simple network examples and potential extensions.The work opens avenues for generalizing to related leakage measures, relaxing key conditions, and deeper characterization of channels achieving bound equality.

Abstract

Maximal leakage quantifies the leakage of information from data due to an observation . 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 . 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 . 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.

Paper Structure

This paper contains 10 sections, 5 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Background and Preliminaries
  4. Simultaneous Coupling for Maximal Leakage
  5. Bounds over Bayesian Networks
  6. Application of Bounds to Simple Bayesian Networks
  7. Minimal Coupling Construction for Leakage Exponent for $n = 4$
  8. Conclusion and Future Directions
  9. Proof of \ref{['prop: Simultaneous Coupling']}
  10. Proof of \ref{['thm: Minimal Coupling for leakage exponent']}

Key Result

Proposition 1

For any random variables $Y_1, \dots, Y_n \in {\mathcal{Y}}$ distributed according to the PMFs $P_{1}, \dots, P_{n} \in {\mathcal{P}}_{\mathcal{Y}}$, respectively, we have where the minimum is taken over all couplings of $P_{1}, \dots, P_{n}$, and ${\mathbb{P}}(\cdot)$ denotes the probability law corresponding to the coupling. Moreover, if then equality in Eq:MaxDoeblinMinimalCoupling holds, whe

Figures (3)

  • Figure 1: Illustrative diagram of a Bayesian network. $X$ is the source node, nodes inside the dotted box belong to the set $V \subset \mathsf{V}$, and $U$ is the sink node.
  • Figure 2:
  • Figure 3:

Theorems & Definitions (6)

  • Proposition 1: Minimal Coupling Bound for Leakage Exponent MakurSingh2024
  • Theorem 1: Simultaneous Coupling
  • Theorem 2: Bounds on Maximal Leakage in Bayesian Networks
  • proof : Proof
  • Corollary 1: Simplified Bounds on Leakage Exponent
  • Theorem 3: Coupling Construction for $n=4$