Suppose that as in Section 1.5 we are interested in
a point process in a region S having unnormalized density 
with respect to a Poisson process with intensity measure 
,
and 
is finite so we have a finite point process. Suppose
that 
with 
, and we see the process
only inside W. We call W the observation window and V the
guard region. We want to do likelihood inference for the parameter
having observed the process in W.
This is a missing data problem. Let x denote the missing data, the process
in V, and y the observed data, the process in W. Let 
denote the complete data, the process in S. In order to carry out MCL
with missing data we need to understand the conditional distribution of x
given y. By the independence properties of the Poisson process and our
slogan from Section 1.3.2, this conditional distribution has
the unnormalized density 
with respect to the
Poisson process on V with intensity measure restricted to V.
Let us again fit the saturation model to the point pattern in
Figure 1.1, but this time we consider the observed pattern to
be a window surrounded by a guard region. Here we take the complete
region to be a torus that is a square of side 1.4 with
edges pasted together. The reason for using a torus is to eliminate the
boundary in the hope of getting something resembling an infinite-volume
process. We do not know whether or not the infinite-volume process exists,
but even if it does we cannot simulate it. Why 1.4? This almost doubles
the area, which in some respects quadruples the work, since we need to
do calculations involving pairs of points. We could try a larger guard
region if we had the patience to do so.
We again start with simulations from the point 
which
was the ``true'' simulation parameter value for the data in
Figure 1.1. We take two samples, one from the unconditional
distribution on S and one for the conditional distribution of the process
on the guard region V given data in the window W, samples of size
with spacing 200. The Monte Carlo MLE was
with MCSE (0.014, 0.0026). If we
compare this with the MCMLE 
with MCSE
(0.0015, 0.00029) that we obtained when we assumed no guard region and
free boundary conditions, we see that the estimates are clearly different.
There is no point in doing a statistical hypothesis test to determine which model, with or without guard region, is correct. In a real application only one model would make scientific sense. Either the process exists outside the observation window or it does not. If the data are positions of trees in a city park surrounded by asphalt, we use a model without a guard region. If the data are positions of trees in a quadrat in a forest, we must use a model with a guard region.
The point of the calculation here is that it makes a big difference which model we use. It is of course well known that it is necessary to deal with boundaries. That is the point of the edge-correction literature. The point of the example here is to show that the same phenomenon arises in likelihood inference. Though this notion of guard regions is very important, being perhaps appropriate for the majority of real applications, we leave it and return to analyses without guard regions.