As an illustration, consider inference about the mean of an exponential distribution with density
The likelihood of a random sample is
in terms of the sufficient statistic3
Take as prior a density of the form
then the corresponding posterior is
which is clearly also of the form (17) and so may be written as
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.