The interest is in inference about given .

With prior

(24) |

, the sample mean, is a sufficient statistic for , as the likelihood, , depends only on and (fixed), through the sampling distribution of which is

(25) | |||

(26) |

so the posterior for is

(27) | |||

(28) |

So, by inspection

(29) |

is a random sample of size from where is known and the prior for is . The posterior is where

(30) |

and

(31) |

The prior for is , and so posterior belief about is as if had a distribution.

danny 2009-07-23