Two alternative exact characterizations of the minimum error probability of Bayesian M-ary hypothesis testing are derived. The first expression corresponds to the error probability of an induced binary hypothesis test and implies the tightness of the meta-converse bound by Polyanskiy et al.; the second expression is a function of an information-spectrum measure and implies the tightness of a generalized Verdú-Han lower bound. The formulas characterize the minimum error probability of several problems in information theory and help to identify the steps where existing converse bounds are loose.

This work was supported in part by the European Research Council under Grant 259663, in part by the Spanish Ministry of Economy and Competitiveness under Grant FPDI-2013-18602, Grant RYC-2011-08150, and Grant TEC2012-38800-C03-03, and in part by the European Union’s 7th Framework Programme under Grant 303633 and Grant 329837.