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7-5

This is recognized as a normal probability density function with posterior mean

2
2
1 n2
0 x
0
2
2

n
0
and posterior variance
2
2
1 1 1 0 1 n2
a
b
2 2 2 2
0 n 0
n
x
and . For purposes of
Consequently, the Bayes estimate of is a weighted average of 0
comparison, note that the maximum likelihood estimate of is ˆ x .
To illustrate, suppose that we have a sample of size n 10 from a normal distribution
2
with unknown mean and variance 4. Assume that the prior distribution for is nor-
2
mal with mean 0 and variance 0 1 . If the sample mean is 0.75, the Bayes estimate
0
of is
14 1020
110.752 0.75
0.536
1
14 102 1.4
Note that the maximum likelihood estimate of is x 0.75 .

There is a relationship between the Bayes estimator for a parameter and the maximum
likelihood estimator of the same parameter. For large sample sizes, the two are nearly
equivalent. In general, the difference between the two estimators is small compared to
1 1n. In practical problems, a moderate sample size will produce approximately the same
estimate by either the Bayes or maximum likelihood method, if the sample results are con-
sistent with the assumed prior information. If the sample results are inconsistent with the
prior assumptions, the Bayes estimate may differ considerably from the maximum likeli-
hood estimate. In these circumstances, if the sample results are accepted as being correct,
the prior information must be incorrect. The maximum likelihood estimate would then be
the better estimate to use.
If the sample results are very different from the prior information, the Bayes estimator
will always tend to produce an estimate that is between the maximum likelihood estimate and
the prior assumptions. If there is more inconsistency between the prior information and the
sample, there will be more difference between the two estimates.

EXERCISES FOR SECTION 7-3.3

S7-1. Suppose that X is a normal random variable (a) Find the posterior distribution for .
2
with unknown mean and known variance . The prior (b) Find the Bayes estimator for .
distribution for is a normal distribution with mean 0 and S7-3. Suppose that X is a Poisson random variable with pa-
2
variance 0 . Show that the Bayes estimator for becomes rameter
. Let the prior distribution for
be a gamma distri-
the maximum likelihood estimator when the sample size n is
bution with parameters m
1 and 1m
12 0 .
large. (a) Find the posterior distribution for
.
S7-2. Suppose that X is a normal random variable with un- (b) Find the Bayes estimator for
.
2
known mean and known variance . The prior distribution
S7-4. Suppose that X is a normal random variable with un-
for is a uniform distribution defined over the interval [a, b]. 2
known mean and known variance 9. The prior distribution