Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

668 Bayesian Statistics

Let us change variables to get more familiar results; namely, let

t=

θ 1 −x

s/

√

n

andθ 1 =x+ts/

√

n,

with Jacobians/

√

n. This conditional pdf oft,givenxands^2 ,isthen

k(t|x, s^2 )∝{(n−1)s^2 +(st)^2 }−n/^2

∝

1

[1 +t^2 /(n−1)][(n−1)+1]/^2

.

That is, the conditional pdf oft=(θ 1 −x)/(s/n), givenxands^2 , is a Studenttwith
n−1 degrees of freedom. Since the mean of this pdf is 0 (assuming thatn>2), it
follows that the Bayes estimator ofθ 1 , under squared-error loss, isX,whichisalso
the mle.
Of course, fromk 1 (θ 1 |x, s^2 )ork(t|x, s^2 ), we can find a credible interval forθ 1.
One way of doing this is to select the highest density region(HDR) of the pdf
θ 1 or that oft. The former is symmetric and unimodal aboutθ 1 and the latter
about zero, but the latter’s critical values are tabulated; so we use the HDR of that
t-distribution. Thus, if we want an interval having probability 1−α,wetake

−tα/ 2 <

θ 1 −x

s/

√

n

<tα/ 2

or, equivalently,

x−tα/ 2 s/

√

n<θ 1 <x+tα/ 2 s/

√

n.

This interval is the same as the confidence interval forθ 1 ; see Example 4.2.1. Hence,
in this case, the improper prior (11.2.1) leads to the same inference as the traditional
analysis.

Example 11.2.2.Usually in a Bayesian analysis, noninformative priors are not

used if prior information exists. Let us consider the same situation as in Example

11.2.1, where the model was aN(θ 1 ,θ 2 ) distribution. Suppose now we consider the

precisionθ 3 =1/θ 2 instead of varianceθ 2. The likelihood becomes

(

θ 3

2 π

)n/ 2

exp

[

−

1

2

{

(n−1)s^2 +n(x−θ 1 )^2

}

θ 3

]

,

so that it is clear that a conjugate prior forθ 3 is Γ(α, β). Further, givenθ 3 ,a
reasonable prior onθ 1 isN(θ 0 ,n 01 θ 3 ), wheren 0 is selected in some way to reflect
how many observations the prior is worth. Thus the joint prior ofθ 1 andθ 3 is

h(θ 1 ,θ 3 )∝θ 3 α−^1 e−θ^3 /β(n 0 θ 3 )^1 /^2 e−(θ^1 −θ^0 )

(^2) θ 3 n 0 / 2
.
If this is multiplied by the likelihood function, we obtain the posterior joint pdf of
θ 1 andθ 3 ,namely,
k(θ 1 ,θ 3 |x, s^2 )∝θ
α+n 2 +^12 − 1
3 exp
[
−
1
2
Q(θ 1 )θ 3
]
,