A simple example that is both a Markov point process and an exponential family
is the Strauss process (Strauss, 1975; Kelly and Ripley, 1976).
Let r be a fixed real number, let 
be a distance
function on S, and let s(x) be the number of pairs of
points in x that are separated by distance no more than r.
Define t(x) to be the bivariate vector (n(x), s(x)).
The exponential family having canonical statistic t(x) and unnormalized
densities (1.3) is called the
unconditional Strauss process.
It was pointed out by Kelly and Ripley (1976) that 
defined
by (1.13) is finite if and only if 
.
Thus the natural parameter space of the family is
Although it is of no importance in the sequel, it is interesting to note
that since 
is not an open subset of 
this is a
nonregular exponential family. The Strauss process thus provides
an interesting example of a nonregular family not dreamed up just to provide
an example of the phenomenon. The nonregularity affects likelihood inference
for the model (Geyer and Møller, 1994). The Strauss process is also
actually nonsteep since 
is finite for all
.
Conditioning on the event 
gives the conditional
Strauss process with k points. As in any exponential family, conditioning
on a component of the natural statistic (here n(x)) eliminates the
corresponding parameter 
. Thus this produces a one-parameter
exponential family with natural statistic s(x). If we do not consider
this a process derived from the unconditional process, but as a simple
multivariate exponential family defined on 
with normalizing function
then is now finite for all 
. Thus this gives
another interesting property not seen in typical exponential families, that
conditioning can increase the natural parameter space.
The title of Strauss (1975) calls this process a `model for clustering,' but the unconditional Strauss process is not at all a model for clustering. By the usual formulas for exponential families
so
and, in particular, every unconditional Strauss process with 
is less clustered than the Poisson process with 
and the same
value of . The conditional Strauss process can be a model for
clustering, since 
is allowed, but it is a very poor model.
Before explaining this we look at another property of exponential families.
