11.4. Modern Bayesian Methods 683
Consider, then, the likelihood function
m(x|γ)=
∫∞
−∞
f(x|θ)h(θ|γ)dθ. (11.4.13)
Using the pdfm(x|γ), we obtain an estimatêγ =̂γ(x), usually by the method
of maximum likelihood. For inference on the parameterθ, the empirical Bayes
procedure uses the posterior pdfk(θ|x,̂γ).
We illustrate the empirical Bayes procedure with the following example.
Example 11.4.3.Consider the same situation discussed in Example 11.4.2, except
assume that we have a random sample onX; i.e., consider the model
Xi|λ, i=1, 2 ,...,n ∼ iid Poisson(λ)
Λ|b ∼ Γ(1,b).
LetX=(X 1 ,X 2 ,...,Xn)′. Hence,
g(x|λ)=
λnx
x 1 !···xn!
e−nλ,
wherex=n−^1
∑n
i=1xi. Thus, the pdf we need to maximize is
m(x|b)=
∫∞
0
g(x|λ)h(λ|b)dλ
=
∫∞
0
1
x 1 !···xn!
λnx+1−^1 e−nλ
1
b
e−λ/bdλ
=
Γ(nx+1)[b/(nb+1)]nx+1
x 1 !···xn!b
.
Taking the partial derivative of logm(x|b) with respect tob,weobtain
∂logm(x|b)
∂b
=−
1
b
+(nx+1)
1
b(bn+1)
.
Setting this equal to 0 and solving forb, we obtain the solution
̂b=x. (11.4.14)
To obtain the empirical Bayes estimate ofλ, we need to compute the posterior pdf
witĥbsubstituted forb. The posterior pdf is
k(λ|x,̂b) ∝ g(x|λ)h(λ|̂b)
∝ λnx+1−^1 e−λ[n+(1/
bb)]
, (11.4.15)
which is the pdf of a Γ(nx+1,̂b/[n̂b+ 1]) distribution. Therefore, the empirical
Bayes estimator under squared-error loss is the mean of this distribution; i.e.,
̂λ=[nx+1]
̂b
n̂b+1
=x, (11.4.16)
sincêb=x. Thus, for the above prior, the empirical Bayes estimate agrees with
the mle.