Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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.