The oldest general class of models specified by unnormalized densities
are exponential families. Let 
be a map
and define
Suppose 
is not the zero measure so that 
for all 
.
Define
For each 
, define
and then define 
by (1.2).
Then 
is
the full exponential family of probability densities with respect
to with canonical statistic t(X), canonical parameter
, and canonical parameter space 
(also called
natural statistic, parameter, and parameter space, respectively).
It is clear that 
is a family of unnormalized densities with respect to specifying
the exponential family 
and that c is the normalizing
function of the family. The term `normalizing function' is not used
in the exponential family literature; c is sometimes called the
Laplace transform of the measure 
, and 
is sometimes called the cumulant function of the family.
These names only apply when the unnormalized densities have the special
structure (1.3), so we shall not use them, being interested
in general families of unnormalized densities.
Of particular interest in spatial statistics and stochastic geometry are exponential families associated with finite-volume Gibbs distributions, Markov random fields, and Markov point processes.
