Operational Interpretation of Renyi Information Measures via Composite Hypothesis Testing Against Product and Markov Distributions

Abstract

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 operational interpretations of various Renyi 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 variations of Renyi mutual information. In case the alternative hypothesis consists of tripartite distributions satisfying the Markov property, we find that the optimal exponents are determined by variations of Renyi conditional mutual information. In either case the relevant notion of Renyi mutual information depends on the precise choice of the alternative hypothesis. As such, our work also strengthens the view that different definitions of Renyi mutual information, conditional entropy and conditional mutual information are adequate depending on the context in which the measures are used.

published version

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• Secure communication over fully quantum Gel'fand-Pinsker wiretap channel
• Universal channel coding for general output alphabet
• Conditional Rényi entropy and the relationships between Rényi capacities
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• Universal tester for multiple independence testing and classical-quantum arbitrarily varying multiple access channel
• On conditional Sibson's $\alpha$-Mutual Information
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• Rényi divergence inequalities via interpolation, with applications to generalised entropic uncertainty relations