### Example

As an illustration, consider inference about the mean of an exponential distribution with density

 (14)

The likelihood of a random sample is

 (15)

in terms of the sufficient statistic3

 (16)

Take as prior a density of the form

 (17) (18) (19)

then the corresponding posterior is

 (20) (21)

which is clearly also of the form (17) and so may be written as

 (22)

where
 (23)

Hence (14) is closed under sampling, with (17) as a convenient natural conjugate prior family for inference about the mean . In such a case a complete analytic solution is available.

danny 2009-07-23