  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.
