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