Applied Statistics and Probability for Engineers

7-5

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

and posterior variance

Consequently, the Bayes estimate of is a weighted average of  0 and. For purposes of

comparison, note that the maximum likelihood estimate of is.

To illustrate, suppose that we have a sample of size n10 from a normal distribution

with unknown mean and variance ^2 4. Assume that the prior distribution for is nor-

mal with mean  0 0 and variance. If the sample mean is 0.75, the Bayes estimate

of is

Note that the maximum likelihood estimate of is.

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

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

(^1)  1 n.
x0.75
(^14)  1020
11 0.75 2
(^1
14)  102

0.75
1.4
0.536
^20  1
ˆx
x
a
1
^20
1
^2 n
b
 1

^20 1 ^2 n 2
^20
^2 n
1 ^2 n 2  0
^20 x
^20
^2 n
S7-1. Suppose that X is a normal random variable
with unknown mean and known variance ^2. The prior
distribution for is a normal distribution with mean  0 and
variance. Show that the Bayes estimator for becomes
the maximum likelihood estimator when the sample size nis
large.
S7-2. Suppose that Xis a normal random variable with un-
known mean and known variance ^2. The prior distribution
for is a uniform distribution defined over the interval [a,b].
^20
(a) Find the posterior distribution for .
(b) Find the Bayes estimator for .
S7-3. Suppose that Xis a Poisson random variable with pa-
rameter. Let the prior distribution for be a gamma distri-
bution with parameters m 1 and.
(a) Find the posterior distribution for.
(b) Find the Bayes estimator for.
S7-4. Suppose that Xis a normal random variable with un-
known mean and known variance ^2 9. The prior distribution
1 m (^12)  0
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