The method of stochastic approximation, used to obtain Monte Carlo maximum likelihood estimates by Younes (1988) and Moyeed and Baddeley (1991) is not useful for obtaining estimates of even moderate precision, nor is there a good method of estimating the accuracy of its estimates. It may be used to get a starting point for MCL methods.
Here we use a very simplified version of stochastic approximation. The
idea is to run a Markov chain with nonstationary transition probabilities,
adjusting the parameter 
in each iteration moving it toward
the MLE. We thus obtain a sequence 
of point patterns
and parameter values. If the current position is 
and x is the
observed point pattern, then a very crude estimate of the score is
. Moving in that direction will, on average, move
it toward the MLE. Hence for some 
, we update using
In classical stochastic approximation, 
is also a function of n,
decreasing with time to that 
converges to the MLE.
The usual practice is to use
in iteration k, where 
and 
are constants.
This form is implemented in the computer code described in
Section 1.13, but since there are no
guidelines for choosing 
or 
, we prefer a more `hands-on'
approach in which 
and 
is chosen by looking
at plots.
There is no known way to choose except by experiment. If
is too large, the sampler may react too rapidly to the initial
position, running away from the MLE. If is too small, the
sampler makes no appreciable progress.
The sampler should remain in approximate equilibrium, so that
is an approximate realization from , and should
be set
small enough so that the sampler stays in approximate equilibrium. One must
not change so fast that the sampler can't change x rapidly enough
to stay in approximate equilibrium.