Statistical Hypothesis Testing and Lower Bounds to the Error Probability

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 Verdú; the second expression is 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. Index Terms Hypothesis testing, meta-converse, information spectrum, channel coding, Shannon theory.