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

TL;DR

This paper provides a comprehensive operational interpretation of Renyi information measures through composite hypothesis testing, analyzing error exponents and asymptotics for product and Markov distribution hypotheses.

Contribution

It introduces a general framework for asymmetric hypothesis testing against composite hypotheses and links Renyi information measures to operational error exponents in these settings.

Findings

Optimal error and strong converse exponents are characterized by Renyi mutual information variations.

Different Renyi measures are appropriate depending on the structure of the alternative hypothesis.

The framework applies to bipartite and tripartite distribution hypotheses with Markov properties.

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.