The range of problems for which an analytic solution is possible is limited and depends on the choice of the model (leading to a tractable likelihood function) and prior.

A simple model assumes that the data is a random sample, of size , from some distribution having density function with

(11) |

If a prior is chosen from some family , say , the choice of being regarded as part of the model specification, the posterior is given by

(12) |

If is also a member of , say

(13) |

where the parameter
is a function of only
and
then the family is said to be *closed
under sampling*, with respect to the density
, (Barnard, 1949). The prior
is called a
*conjugate prior* for
(see Smith and Bernardo, 1994, p. 265).