  Exponential families of discrete probability distributions when the
normalizing constant (or overall effect) is added or removed are compared in
this paper. The latter setup, in which the exponential family is curved, is
particularly relevant when the sample space is an incomplete Cartesian product
or when it is very large, so that the computational burden is significant. The
lack or presence of the overall effect has a fundamental impact on the
properties of the exponential family. When the overall effect is added, the
family becomes the smallest regular exponential family containing the curved
one. The procedure is related to the homogenization of an inhomogeneous variety
discussed in algebraic geometry, of which a statistical interpretation is given
as an augmentation of the sample space. The changes in the kernel basis
representation when the overall effect is included or removed are derived. The
geometry of maximum likelihood estimates, also allowing zero observed
frequencies, is described with and without the overall effect, and various
algorithms are compared. The importance of the results is illustrated by an
example from cell biology, showing that routinely including the overall effect
leads to estimates which are not in the model intended by the researchers.
