Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

664 Bayesian Statistics

0 5 10 15 20

0.00

0.02

0.04

0.06

0.08

0.10

θ

h(

θ)

0 5 10 15 20

0.0

0.1

0.2

0.3

0.4

0.5

0.6

θ

k(θ

|x)

Figure 11.1.1:Prior (left panel) and posterior (right panel) pdfs of Example 11.1.6

ThusP[Θ> 10 |y= 177] = 1− 0 .9368 = 0.0632; consequently, our rule would accept

H 0.

The 95% credible interval, (11.1.12), is (7. 77 , 10 .31), which also contains 10; see

Exercise 11.1.7 for details.

11.1.5 BayesianSequentialProcedures.................

Finally, we should observe what a Bayesian would do if additional data were col-
lected beyondx 1 ,x 2 ,...,xn. In such a situation, the posterior distribution found
with the observationsx 1 ,x 2 ,...,xnbecomes the new prior distribution, additional
observations give a new posterior distribution, and inferences would be made from
that second posterior. Of course, this can continue with even more observations.
That is, the second posterior becomes the new prior, and the next set of observa-
tions yields the next posterior from which the inferences can be made. Clearly, this
gives Bayesians an excellent way of handling sequential analysis. They can continue
taking data, always updating the previous posterior, which has become a new prior
distribution. Everything a Bayesian needs for inferences is in that final posterior
distribution obtained by this sequential procedure.

EXERCISES

11.1.1.LetYhave a binomial distribution in whichn=20andp=θ. The prior

probabilities onθareP(θ=0.3) = 2/3andP(θ=0.5) = 1/3. Ify=9,whatare

the posterior probabilities forθ=0.3andθ=0.5?