In the notation of Section 3.2, consider the problem of testing H0 : P = P0 against H1 : P = P1, and suppose that known probabilities π0 = π

and π1 = 1 − π can be assigned to H0 and H1 prior to the experiment.

(i) The overall probability of an error resulting from the use of a test ϕ is

πE0ϕ(X) + (1 − π)E1[1 − ϕ(X)].

(ii) The Bayes test minimizing this probability is given by (3.8) with k =

π0/π1.

(iii) The conditional probability of Hi given X = x, the posterior probability of Hi is

πipi(x)

π0p0(x) + π1p1(x)

, and the Bayes test therefore decides in favor of the hypothesis with the larger posterior probability