Suppose now that we are given a family of normalized probability densities
but are interested in the conditional family
.
What this means is the following. Say 
is a density with
respect to 
. For fixed y consider the function
This is an unnormalized density with respect to 
with normalizing
constant
where 
is the marginal density with respect to 
of the
random variable Y. The normalized density is then
Conditional families arise in conditional likelihood inference and in
Bayesian inference. In conditional likelihood inference, although
the complete data (x, y) are observed, rather than maximizing the
joint likelihood
to estimate 
, one maximizes the conditional likelihood
. Thus is usually done because because y is exactly
or approximately ancillary.
In Bayesian inference the likelihood times the prior is a known normalized
joint density for parameter and data, but the object of inference is the
conditional distribution of the parameter given the data.
