Operational interpretation of Rényi information measures via composite hypothesis testing against product and markov distributions

© 1963-2012 IEEE. We revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis. We introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms. In this case, we find the threshold rate, the optimal error and strong converse exponents (at large deviations from the threshold), and the second order asymptotics (at small deviations from the threshold). We apply our results to find the operational interpretations of various Rényi information measures. In case the alternative hypothesis is comprised of bipartite product distributions, we find that the optimal error and strong converse exponents are determined by the variations of Rényi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by the variations of Rényi conditional mutual information. In either case, the relevant notion of Rényi mutual information depends on the precise choice of the alternative hypothesis. As such, this paper also strengthens the view that different definitions of Rényi mutual information, conditional entropy, and conditional mutual information are adequate depending on the context in which the measures are used

Hypothesis Testing Interpretations and Renyi Differential Privacy

Differential privacy is a de facto standard in data privacy, with applications in the public and private sectors. A way to explain differential privacy, which is particularly appealing to statistician and social scientists is by means of its statistical hypothesis testing interpretation. Informally, one cannot effectively test whether a specific individual has contributed her data by observing the output of a private mechanism---any test cannot have both high significance and high power. In this paper, we identify some conditions under which a privacy definition given in terms of a statistical divergence satisfies a similar interpretation. These conditions are useful to analyze the distinguishability power of divergences and we use them to study the hypothesis testing interpretation of some relaxations of differential privacy based on Renyi divergence. This analysis also results in an improved conversion rule between these definitions and differential privacy

Information Theoretic Structure Learning with Confidence

Information theoretic measures (e.g. the Kullback Liebler divergence and Shannon mutual information) have been used for exploring possibly nonlinear multivariate dependencies in high dimension. If these dependencies are assumed to follow a Markov factor graph model, this exploration process is called structure discovery. For discrete-valued samples, estimates of the information divergence over the parametric class of multinomial models lead to structure discovery methods whose mean squared error achieves parametric convergence rates as the sample size grows. However, a naive application of this method to continuous nonparametric multivariate models converges much more slowly. In this paper we introduce a new method for nonparametric structure discovery that uses weighted ensemble divergence estimators that achieve parametric convergence rates and obey an asymptotic central limit theorem that facilitates hypothesis testing and other types of statistical validation.Comment: 10 pages, 3 figure