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Two alternative exact characterizations of the minimum error probability ofBayesian M-ary hypothesis testing are derived. The first expression correspondsto the error probability of an induced binary hypothesis test and implies thetightness of the meta-converse bound by Polyanskiy, Poor and Verd\'u; thesecond expression is function of an information-spectrum measure and impliesthe tightness of a generalized Verd\'u-Han lower bound. The formulascharacterize the minimum error probability of several problems in informationtheory and help to identify the steps where existing converse bounds are loose.

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