Finite point processes are easily simulated by a special case of the
MHG algorithm described by Geyer and Møller (1994). Let h be
the unnormalized density with respect to 
of the process.
Suppose the points are ordered, as they must be to be stored in a computer,
If n(x) = m write 
.
A proposal kernel 
defined as follows, when n(x) = m
propose to add a point 
distributed on S with distribution
proportional to 
, when n(x) = m + 1 propose to delete
. For 
this kernel does nothing
( 
).
First consider the part of the proposal that attempts to add a point.
Then x is in 
, and the proposal y is in 
The new point 
is distributed on S with distribution proportional
to and the rest of the points are not moved,
for 
. Thus the joint distribution of the pair (x,y)
is concentrated on the set
This set is not symmetric. Interchange of x and y gives
is isomorphic to via the map 
, and
is isomorphic to via the map 
.
Let 
be the measure on 
concentrated on
and `equal' to 
on and ,
where the inverted commas call attention to the just-mentioned isomorphism.
Still considering a proposed addition of a point with 
.
the unnormalized joint density of (x,y) with respect to is
the factor 
arising from the proposal being
from the probability measure 
.
Now considering a proposed deletion of with 
we obtain
This now allows us to calculate Green's ratio. For a proposal going up Green's ratio is (1.18) with x and y interchanged divided by (1.17)
For a proposal going down, Green's ratio is the reciprocal.
Now consider composing this update mechanism with an update that merely
permutes the order of the points. Then at each update step
is a random one of the m + 1 points. Hence we use the same Green's
ratio whether we always delete the last point or whether we delete a
random point. Let 
be the kernel that performs the update
just described, proposing a random point to delete.
Finally consider the mixture
This proposes half the time to add a point and half the time to delete a point,
except when 
in which case it is impossible to go down. When
, this proposes half the time to add a point and half the time
to do nothing.
In summary the algorithm is
