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

The likelihood of a random sample is

(15) |

in terms of the sufficient statistic^{3}

(16) |

Take as prior a density of the form

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