Missing data involve the same considerations as conditional families.
If x is missing and y is observed and the joint density is
, then the likelihood is the marginal density
considered a function of
for y fixed at the observed value.
As we observed in the preceding section
is the normalizing
constant for the joint density considered as an unnormalized conditional
density (1.4). Thus the family of unnormalized densities is
involved in both conditional likelihood inference and likelihood inference
with missing data.
Latent variables, random effects, and ordinary (non-Bayes) empirical Bayes models all involve missing data of some form, and all give rise to the same considerations.