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A Generalized Leakage Interpretation of Alpha-Mutual Information

Akira Kamatsuka, Takahiro Yoshida

TL;DR

The paper addresses how $α$-mutual information ($α$-MI) can be interpreted within a generalized leakage framework. It develops a KN-mean–based generalized vulnerability and a $q$-logarithmic generalization of Gibbs' inequality to express $α$-MI as generalized $g$-leakage, unifying several $α$-MI variants under a single mechanism. It shows that the parameter $α$ reflects the adversary’s risk aversion, providing a utility-theoretic interpretation via the transformed gain $g_{α}(x,r)=\ln_{1/α} r(x)$ and the Arrow–Pratt measure $A_{g_{α}}(t)=1/(α t)$, with extensions beyond the Arimoto case to Sibson, Hayashi, and LP forms. This framework links information-theoretic leakage concepts with decision-theoretic risk preferences, offering a cohesive, interpretable view of information leakage across $α$-MI measures.

Abstract

This paper presents a unified interpretation of $α$-mutual information ($α$-MI) in terms of generalized $g$-leakage. Specifically, we present a novel interpretation of $α$-MI within an extended framework for quantitative information flow based on adversarial generalized decision problems. This framework employs the Kolmogorov-Nagumo mean and the $q$-logarithm to characterize adversarial gain. Furthermore, we demonstrate that, within this framework, the parameter $α$ can be interpreted as a measure of the adversary's risk aversion.

A Generalized Leakage Interpretation of Alpha-Mutual Information

TL;DR

The paper addresses how -mutual information (-MI) can be interpreted within a generalized leakage framework. It develops a KN-mean–based generalized vulnerability and a -logarithmic generalization of Gibbs' inequality to express -MI as generalized -leakage, unifying several -MI variants under a single mechanism. It shows that the parameter reflects the adversary’s risk aversion, providing a utility-theoretic interpretation via the transformed gain and the Arrow–Pratt measure , with extensions beyond the Arimoto case to Sibson, Hayashi, and LP forms. This framework links information-theoretic leakage concepts with decision-theoretic risk preferences, offering a cohesive, interpretable view of information leakage across -MI measures.

Abstract

This paper presents a unified interpretation of -mutual information (-MI) in terms of generalized -leakage. Specifically, we present a novel interpretation of -MI within an extended framework for quantitative information flow based on adversarial generalized decision problems. This framework employs the Kolmogorov-Nagumo mean and the -logarithm to characterize adversarial gain. Furthermore, we demonstrate that, within this framework, the parameter can be interpreted as a measure of the adversary's risk aversion.
Paper Structure (11 sections, 7 theorems, 35 equations, 1 figure)

This paper contains 11 sections, 7 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\alpha$-Mutual information ($\alpha$-MI)
  4. $q$-Logarithm and Kolmogorov--Nagumo (KN) Mean
  5. Generalized Vulnerability and Generalized $g$-Leakage
  6. Main Results
  7. Generalized $g$-Leakage expressions for $\alpha$-MI
  8. An Interpretation of $\alpha$ as the Adversary's Risk Aversion
  9. Conclusion
  10. Proof of Proposition \ref{['prop:generalized_Gibbs_inequality']}
  11. Proof of Theorem \ref{['thm:alpha_conditional_entropy_vulnerability']}

Key Result

Proposition 1

For $\alpha\in (0, 1)\cup (1, \infty)$, where and $p_{X_{\frac{1}{\alpha}}} = p_{X}^{(\frac{1}{\alpha})}$, the $\frac{1}{\alpha}$-tilted distribution of $p_{X}$. For $\alpha\in (1/2, 1) \cup (1, \infty)$, where and $p_{X_{\frac{\alpha}{2\alpha-1}}} = p_{X}^{(\frac{\alpha}{2\alpha-1})}$, the $\frac{\alpha}{2\alpha-1}$-tilted distribution of $p_{X}$.

Figures (1)

  • Figure 1: Plot of $g_{\alpha}(x, r)=\ln_{1/\alpha} r(x)$

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1: 11195284
  • Definition 3: $q$-logarithm and $q$-exponential
  • Remark 2
  • Proposition 2: Generalized Gibbs' inequality
  • proof
  • Remark 3
  • Definition 4: Kolmogorov--Nagumo (KN) mean
  • ...and 18 more