662 Bayesian Statistics
11.1.3 BayesianIntervalEstimation
If an interval estimate ofθis desired, we can find two functionsu(x)andv(x)so
that the conditional probability
P[u(x)<Θ<v(x)|X=x]=
∫v(x)
u(x)
k(θ|x)dθ
is large, for example, 0.95. Then the intervalu(x)tov(x)isanintervalestimate
ofθin the sense that the conditional probability of Θ belonging to that interval is
equal to 0.95. These intervals are often calledcredibleorprobability intervals,
so as not to confuse them with confidence intervals.
Example 11.1.4. Consider Example 11.1.3, whereX 1 ,X 2 ,...,Xnis a random
sample from aN(θ, σ^2 ) distribution, whereσ^2 is known, and the prior distribution
is a normalN(θ 0 ,σ 02 ) distribution. The statisticY =Xis sufficient. Recall that
the posterior pdf of Θ givenY=ywas normal with mean and variance given near
expression (11.1.11). Hence a credible interval is found by taking the mean of the
posterior distribution and adding and subtracting 1.96 of its standard deviation;
that is, the interval
yσ 02 +θ 0 (σ^2 /n)
σ^20 +(σ^2 /n)
± 1. 96
√
(σ^2 /n)σ^20
σ 02 +(σ^2 /n)
forms a credible interval of probability 0.95 forθ.
Example 11.1.5.Recall Example 11.1.1, whereX′=(X 1 ,X 2 ,...,Xn) is a random
sample from a Poisson distribution with meanθand a Γ(α, β)prior,withαand
βknown, is considered. As given by expression (11.1.7), the posterior pdf is a
Γ(y+α, β/(nβ+1)) pdf, wherey=
∑n
i=1xi. Hence, if we use the squared-error
loss function, the Bayes point estimate ofθis the mean of the posterior
δ(y)=
β(y+α)
nβ+1
=
nβ
nβ+1
y
n
+
αβ
nβ+1
.
As with the other Bayes estimates we have discussed in this section, for largen
this estimate is close to the maximum likelihood estimate and the statisticδ(Y)
is a consistent estimate ofθ. To obtain a credible interval, note that the posterior
distribution of2(nββ+1)Θisχ^2 (2(
∑n
i=1xi+α)). Based on this, the following interval
is a (1−α)100% credible interval forθ:
(
β
2(nβ+1)
χ^21 −(α/2)
[
2
(n
∑
i=1
xi+α
)]
,
β
2(nβ+1)
χ^2 α/ 2
[
2
(n
∑
i=1
xi+α
)])
,
(11.1.12)
whereχ^21 −(α/2)(2(
∑n
i=1xi+α)) andχ
2
α/ 2 (2(
∑n
i=1xi+α)) are the lower and upper
χ^2 quantiles for aχ^2 distribution with 2(
∑n
i=1xi+α) degrees of freedom.