An Algorithmic Description - Design and Analysis of Learning Classifier Systems - page 197

Thus, the frequency of classifier type C i only remains unchanged if there is no
such classifier in the current population, or if its fitness equals the average fitness
of the current population. The population is stable only if this applies to all its
classifiers.
One wants to define a fitness function for each classifier such that the sta-
ble population is the optimal population according to the optimality criterion.
Currently

( q ) by (7.96) cannot be fully split into one component per classifier
due to the term ln | Λ V − 1

L

| in L M ( q ) that results from the mixing model. Re-
placing this mixing model by heuristics should make such a split possible. Even
then it is for each classifier a function of all classifiers in the current population,
as the mixing coecients assigned to a single classifier for some input depend
on other classifiers that match the same input, which conforms to the above
definition of the fitness of a classifier type being a function of the frequency of
all classifier types.
The stable state of the population is given if a classifier's fitness is equal to
the average fitness of all classifiers. This seems very unlikely to result naturally
from splitting ( q ) into the classifier components, and thus either (8.15) needs

to be modified, or the fitness function needs to be tuned so that this is the case.
If and how this can be done cannot be answered before the fitness function is
available. Furthermore, (8.15) does not allow the emergence of classifiers that
initially have a frequency of 0. As initialising the population with all possible
classifiers is not feasible even for rather small problems, new classifier types need
to be added stochastically and periodically. To make this possible, (8.15) needs
to be modified to take this into account, resulting in a stochastic equation.
Obviously, a lot more work is required to see if the replicator dynamics ap-
proach can be used to design Michigan-style LCS. If it can, the approach opens
the door to applying the numerous tools designed to analyse replicator dynamics
to the analysis of the classifier dynamics in Michigan-style LCS.

L

8.5 Summary

In this chapter it was demonstrated how to the optimality criterion that was
introduced in the previous chapter can be applied by implementing variational
Bayesian inference together with some model structure search procedure. Four
simple regression tasks were used to demonstrate that the optimality criterion
based on model selection yields adequate results.
A set of function were provided that perform variational Bayesian inference to
approximate the model probability p (

) and act as a basis for evaluating the
quality of a set of classifiers. More specifically, the function ModelProbability
takes the model structure

M|D

as arguments and returns an
approximation to the unnormalised model probability. Thus, in addition to the
theoretical treatment of variational inference in the previous chapter, it was
shown here how to implement it for the regression case. Due to required complex
procedure of finding the mixing weight vectors to combine the localised classifier
models to a global model, the described implementation scales unfavourably

M and the data D