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.

This study presents alternating optimization (AO) algorithms for computing$\alpha$-mutual information ($\alpha$-MI) and$\alpha$capacity based on variational characterizations of$\alpha$-MI using a reverse channel. Specifically, we derive several variational characterizations of Sibson, Arimoto, Augustin-Csiszár, and LapidothPfister MI and introduce novel AO algorithms for computing$\alpha$MI and$\alpha$-capacity; their performances for computing$\alpha$-capacity are also compared. The comparison results show that the AO algorithm based on the Sibson MI's characterization has the fastest convergence speed. A full version [1] with all proofs, explanations and more discussions is accessible at: https://arxiv.org/abs/2404.10950

In this paper, we present several novel representations of$\alpha$-mutual information ($\alpha$-MI) in terms of Rényi divergence and conditional Rényi entropy. The representations are based on the variational characterizations of$\alpha$-MI using a reverse channel. Based on these representations, we provide several interpretations of the$\alpha-\text{MI}$as privacy leakage measures using generalized mean and gain functions. Further, as byproducts of the representations, we propose novel conditional Rényi entropies that satisfy the property that conditioning reduces entropy and data-processing inequality.

The Sibson and Arimoto capacity, which are based on the Sibson and Arimoto mutual information (MI) of order α, respectively, are well-known generalizations of the channel capacity C. In this study, we derive novel alternating optimization algorithms for computing these capacities by providing new variational characterizations of the Sibson and Arimoto MI. Moreover, we prove that all iterative algorithms for computing these capacities are equivalent under appropriate conditions imposed on their initial distributions.

The problem of computing$\alpha$-capacity for$\alpha > 1$is equivalent to that of computing the correct decoding exponent. Various algorithms for computing them have been proposed, such as Arimoto and Jitsumatsu-Oohama algorithm. In this study, we propose a novel alternating optimization algorithm for computing the$\alpha$-capacity for$\alpha > 1$based on a variational characterization of the Augustin-Csiszár mutual information. A comparison of the convergence performance of these algorithms is demonstrated through numerical examples.

The Stratonovich’s value of information (VoI) is quantity that measures how much inferential gain is obtained from noisy data under information leakage constraint. In this paper, we introduce a generalized VoI for a general loss function and general information leakage. Then we derive an upper bound of the generalized VoI. Moreover, for a classical loss function, we provide a achievable condition of the upper bound which is weaker than that of in previous studies. Since VoI can be viewed as a formulation of a privacy-utility trade-off (PUT) problem, we provide an interpretation of the achievable condition in the PUT context.

The hierarchical and recursive expressive capability of rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. On the other hand, such hierarchical expressive capability causes a problem in tree selection to avoid overfitting. One unified approach to solve this is a Bayesian approach, on which the rooted tree is regarded as a random variable and a direct loss function can be assumed on the selected model or the predicted value for a new data point. However, all the previous studies on this approach are based on the probability distribution on full trees, to the best of our knowledge. In this paper, we propose a generalized probability distribution for any rooted trees in which only the maximum number of child nodes and the maximum depth are fixed. Furthermore, we derive recursive methods to evaluate the characteristics of the probability distribution without any approximations.

We consider the problem of publishing data with utility and privacy guarantees in a statistical decision-theoretical framework. In this framework, we introduce a statistical decision-theoretic quantity called average gain for measuring not only privacy but also utility. We also show a relationship between the average gain and the $\alpha$-leakage, a tunable leakage measure proposed by Liao et at. Moreover, we formulate the privacyutility trade-off (PUT) problem using Stratonovich’s value of information (VoI) and present an analysis of the PUT.