Page 514 - Probability and Statistical Inference

10. Bayesian Methods 491

one can immediately write down the HPD 100(1 – α)% credible interval Θ*
for : Let us denote

with the largest positive number a such that P{Θ* |T = t) ≥ 1 – α. !
Example 10.5.4 (Example 10.5.3 Continued) In order to gather infor-
mation about a new pain-killer, it was administered to n = 10 comparable
patients suffering from the same type of headache. Each patient was checked
after one-half hour to see if the pain was gone and we found seven patients
reporting that their pain vanished. It is postulated that we have observed the
random variables X , ..., X which are iid Bernoulli(θ) given that = θ
1 10
where 0 < < 1 is the unknown probability of pain relief for each patient, 0
< < 1. Suppose that the prior distribution of was fixed in advance of
data collection as Beta(2, 6). Now, we have observed seven patients who
got the pain relief. From (10.5.5), one can immediately write down the form
of the HPD 95% credible interval Θ* for as

with the largest positive number a such that P{Θ* |T = t) = 1 – α. Let us
rewrite (10.5.6) as

We will choose the positive number b in (10.5.7) such that P{Θ*(b) |T = t) =
1 – α. We can simplify (10.5.7) and equivalently express, for 0 < b < ¼,

Now, the posterior probability content of the credible interval Θ*(b) is given
by

We have given a plot of the function q(b) in the Figure 10.5.1. From this
figure, we can guess that the posterior probability of the credible interval