Bayesian M-ary Hypothesis Testing: The Meta-Converse and Verd\'u-Han Bounds are Tight

Abstract

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, Poor and Verd\'u; the second expression is function of an information-spectrum measure and implies the tightness of a generalized Verd\'u-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.