  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.
