The discrepancies between p-values and Bayes posterior probabilities are not as dramatic in the one-sided problem, as is discussed by Casella and Berger (1987) and also mentioned in the Miscellanea section. Let X1,...,Xn be a random sample from a n(θ, H0: θ ‰¤ 0 versus H1: θ > 0.
The prior distribution on θ is n(0, Ï„2), Ï„2 known, which is symmetric about the hypotheses in the sense that P(θ ‰¤ 0) = P(θ > 0) = 1/2.
(a) Calculate the posterior probability that H0 is true, P(θ (b) Find an expression for the p-value corresponding to a value of x, using tests that reject for large values of X.
(c) For the special case σ2 = Ï„2 = 1, compare P(θ ‰¤ 0|x1,... ,xn) and the p-value for values of x > 0. Show that the Bayes probability is always greater than the p-value.
(d) Using the expression derived in parts (a) and (b), show that

an equality that does not occur in the two-sided problem.

lim P(9-0|z , , , ., zn)=p-value,