11.1. Bayesian Procedures 657Example 11.1.1.Consider the modelXi|θ ∼ iid Poisson(θ)
Θ ∼ Γ(α, β),αandβare known.Hence the random sample is drawn from a Poisson distribution with meanθand
the prior distribution is a Γ(α, β) distribution. LetX′=(X 1 ,X 2 ,...,Xn). Thus,
in this case, the joint conditional pdf ofX,givenΘ=θ, (11.1.2), isL(x|θ)=
θx^1 e−θ
x 1!···
θxne−θ
xn!,xi=0, 1 , 2 ,... ,i=1, 2 ,...,n,and the prior pdf ish(θ)=
θα−^1 e−θ/β
Γ(α)βα, 0 <θ<∞.Hence the joint mixed continuous-discrete pdf is given byg(x,θ)=L(x|θ)h(θ)=[
θx^1 e−θ
x 1!
···θxne−θ
xn!][
θα−^1 e−θ/β
Γ(α)βα]
,provided thatxi=0, 1 , 2 , 3 ,..., i=1, 2 ,...,n,and0<θ<∞, and is equal to
zero elsewhere. Then the marginal distribution of the sample, (11.1.4), isg 1 (x)=∫∞0θPx
i+α−^1 e−(n+1/β)θ
x 1 !···xn!Γ(α)βαdθ=Γ(n
∑1xi+α)x 1 !···xn!Γ(α)βα(n+1/β)Px
i+α.
(11.1.6)
Finally, the posterior pdf of Θ, givenX=x, (11.1.5), is
k(θ|x)=L(x|θ)h(θ)
g 1 (x)=θPxi+α− (^1) e−θ/[β/(nβ+1)]
Γ
(∑
xi+α
)
[β/(nβ+1)]
Px
i+α
, (11.1.7)
provided that 0<θ<∞, and is equal to zero elsewhere. This conditional pdf is of
the gamma type, with parametersα∗=
∑n
i=1xi+αandβ
∗=β/(nβ+ 1). Notice
that the posterior pdf reflects both prior information (α, β) and sample information
(
∑n
i=1xi).
In Example 11.1.1, notice that it is not really necessary to determine the marginal
pdfg 1 (x) to find the posterior pdfk(θ|x). If we divideL(x|θ)h(θ)byg 1 (x), we
must get the product of a factor that depends uponxbut doesnotdepend uponθ,
sayc(x), and
θ
Px
i+α−^1 e−θ/[β/(nβ+1)].
That is,
k(θ|x)=c(x)θ
Px
i+α−^1 e−θ/[β/(nβ+1)],