Suppose that we wish to test the hypotheses (9.8.1). Let the loss function have the form of (9.8.2).
a. Prove that the posterior probability of θ = θ0 is ξ0f0(x)/[ξ0f0(x) + ξ1f1(x)].
b. Prove that a test that minimizes r(δ) also minimizes the posterior expected value of the loss given X = x for all x.
c. Prove that the following test is one of the tests described in part (b): “reject H0 if Pr(H0 true|x) ≤ w1/(w0 + w1).”