11.2. More Bayesian Terminology and Ideas 669whereQ(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 )∝∫∞0k(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 lettingt=θ 1 −n^0 nθ 00 ++nnx
√
D/[(n 0 +n)(2α+n)],with Jacobian
√
D/[(n 0 +n)(2α+n)]. Thusk 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 +nxk=n 0 θ 0 +nx
n 0 +ns^2 k=D
2 α+n,thent=θ 1 −xk
sk/
√
nk