Page 528 - Probability and Statistical Inference

10. Bayesian Methods 505

θ given that = θ.
(i) Derive the marginal pdf m(t) of T for t ∈ ℜ;
(ii) Derive the posterior pdf k(θ t) for v given that T = t, for θ > 0 and
–∞ < t < ∞;
(iii) Under the squared error loss function, derive the Bayes estimate
and show, as in the Example 10.7.3, that can take only positive
values.
{Hint: Proceed along the Example 10.7.4.}
10.7.6 (Exercise 10.7.5 Continued) Let X , ..., X be iid N (θ, 1) given that
1
10
= θ where (∈ ℜ) is the unknown parameter. Let us suppose that the prior
pdf of on the space Θ = ℜ is given by h(θ) = ½αe –α|θ| I (θ ∈ ℜ) where α(>
0) is a known number. Suppose that the observed value of the sample mean
was 5.92. Plot the posterior pdf k(θ t) assigning the values α = ½, ¼ and 1/8.
Comment on the empirical behaviors of the corresponding posterior pdf and
the Bayes estimates of .