In Example 8.2.7 we saw an example of a one-sided Bayesian hypothesis test. Now we will consider a similar situation, but with a two-sided test. We want to test
H0: θ = 0 versus H1: θ ≠ 0,
and we observe X1,... ,Xn, a random sample from a n(θ, σ2) population, σ2 known.
A type of prior distribution that is often used in this situation is a mixture of a point mass on θ = 0 and a pdf spread out over H1. A typical choice is to take P(θ = 0) = 5, and if θ = 0, take the prior distribution to be 1/2(0, τ2), where τ2 is known.
(a) Show that the prior defined above is proper, that is, P(-∞ < θ < ∞) = 1.
(b) Calculate the posterior probability that Ho is true, P{θ = 0|x1,..., xn).
(c) Find an expression for the p-value corresponding to a value of x.
(d) For the special case σ2 = τ2 = 1, compare P(θ = 0|x1,... ,xn) and the p-value for a range of values of x. In particular,
(i) For n = 9, plot the p-value and posterior probability as a function of x, and show that the Bayes probability is greater than the p-value for moderately large values of x.
(ii) Now, for a = .05, set x = Zα/2/√n, fixing the p-value at a for all n. Show that the posterior probability at x = Za/2/√n goes to 1 as n → ∞. This is Lindley's Paradox.
Small values of P(θ = 0|x1,... ,xn) are evidence against H0, and thus this quantity is similar in spirit to a p-value. The fact that these two quantities can have very different values was noted by Lindley (1957) and is also examined by Berger and Sellke (1987). (See the Miscellanea section.)