11.2. More Bayesian Terminology and Ideas 669
where
Q(θ 1 )=
2
β
+n 0 (θ 1 −θ 0 )^2 +[(n−1)s^2 +n(x−θ 1 )^2 ]
=(n 0 +n)
[(
θ 1 −
n 0 θ 0 +nx
n 0 +n
) 2 ]
+D,
with
D=
2
β
+(n−1)s^2 +(n− 01 +n−^1 )−^1 (θ 0 −x)^2.
If we integrate outθ 3 ,weobtain
k 1 (θ 1 |x, s^2 )∝
∫∞
0
k(θ 1 ,θ 3 |x, s^2 )dθ 3
∝
1
[Q(θ 1 )][2α+n+1]/^2
.
To get this in a more familiar form, change variables by letting
t=
θ 1 −n^0 nθ 00 ++nnx
√
D/[(n 0 +n)(2α+n)]
,
with Jacobian
√
D/[(n 0 +n)(2α+n)]. Thus
k 2 (t|x, s^2 )∝
1
[
1+ 2 αt+^2 n
](2α+n+1)/ 2 ,
which is a Studenttdistribution with 2α+ndegrees of freedom. The Bayes estimate
(under squared-error loss) in this case is
n 0 θ 0 +nx
n 0 +n
.
It is interesting to note that if we define “new” sample characteristics as
nk=n 0 +n
xk=
n 0 θ 0 +nx
n 0 +n
s^2 k=
D
2 α+n
,
then
t=
θ 1 −xk
sk/
√
nk