Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

672 Bayesian Statistics

11.2.6.Consider the Bayes model

Xi|θ,i=1, 2 ,...,n ∼ iid with distribution Poisson (θ),θ> 0

Θ ∼ h(θ)∝θ−^1 /^2.

(a)Show thath(θ)isintheclassofJeffreys’priors.

(b)Show that the posterior pdf of 2nθis the pdf of aχ^2 (2y+ 1) distribution,

wherey=

∑n

i=1xi.

(c)Use the posterior pdf of part (b) to obtain a (1−α)100% credible interval for

θ.

(d)Use the posterior pdf in part (d) to determine a Bayesian test for the hypothe-

sesH 0 :θ≥θ 0 versusH 1 :θ<θ 0 ,whereθ 0 is specified.

11.2.7.Consider the Bayes model

Xi|θ,i=1, 2 ,...,n∼iid with distributionb(1,θ), 0 <θ< 1.

(a)Obtain the Jeffreys’ prior for this model.

(b)Assume squared-error loss and obtain the Bayes estimate ofθ.

11.2.8.Consider the Bayes model

Xi|θ,i=1, 2 ,...,n ∼ iid with distributionb(1,θ), 0 <θ< 1

Θ ∼ h(θ)=1.

(a)Obtain the posterior pdf.

(b)Assume squared-error loss and obtain the Bayes estimate ofθ.

11.2.9.LetX 1 ,X 2 ,...,Xnbe a random sample from a multivariate normal normal
distributionwithmeanvectorμ=(μ 1 ,μ 2 ,...,μk)′and known positive definite
covariance matrixΣ.LetXbe the mean vector of the random sample. Suppose
thatμhas a prior multivariate normal distribution with meanμ 0 and positive
definite covariance matrixΣ 0. Find the posterior distribution ofμ,givenX=x.
Then find the Bayes estimateE(μ|X=x).

11.3GibbsSampler

From the preceding sections, it is clear that integration techniques play a significant

role in Bayesian inference. Hence, we now touch on some of the Monte Carlo

techniques used for integration in Bayesian inference.

The Monte Carlo techniques discussed in Chapter 5 can often be used to ob-

tain Bayesian estimates. For example, suppose a random sample is drawn from a