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Mathematical models and methods 571
This model is of interest for the following reasons. First of all instead of
seeking to plot a learning curve direct, the researcher tries to represent the
learning process by a physical model and this model has the advantage of
taking into account an obviously essential parameter: the 0 parameter, which
measures the effect of reinforcement. Further, the model is so constructed as to
eliminate the quantity s (number of elements) in the equation giving rz(n) as a
function of n and 0. Finally the model has thevirtue of providing a basic schema
which can be elaborated to adapt it to other experimental situations.
In point of fact, the variants of this model are numerous. Thus experiments
have been undertaken where the frequency of weighting conditions depends on
the subject's responses : here the experimenter chooses EI with probability nII
and E, with probability 7c12 if the subject has chosen A,. Again the model has
been extended by Estes, Suppes and Atkinson to handle social interaction
situations.45 By way of example let us look at an experiment summarized by the
following matrix.
A2
B
Here the two subjects are unaware of each other's responses, but the response
of the experimenter depends on the combination of the two subjects' responses.
If A chooses AI and B chooses BI, A's response is given a probability a, and
B's a probability of (I -a,). The other cells of the matrix are to be interpreted
in a like manner. An analysis on the same lines as that of the Estes and Burke
model shows that one can express this situation by a transition matrix containing
such items as PZ1,,,
(probability of passing from a situation where A is in state
2 and B in state I to a situation where A is in state I and B in state 2). As before,
the learning curves can be deduced from this.
Though there is no question of reviewing, however summarily, the complete
methodology used in the theory of learning, be it noted that the 'sampling
theory' we have been discussing in the preceding paragraphs is only one segment
of the full range of models applied to learning. As well as the sampling theory,
there is a whole school of research characterized by procedures consisting basi-
cally in recourse to an operator to arrive ar r("). If we call this operator Q,,
r(") is given by an equation of the form:
= Qrjr("
- I)
The operator Q, is determined by the responses (AI) and by the investigator's
interventions (Ej). The stimulus is always the same, as in the situations dealt
with by the theory of sampling, while the operators QiJ may be chosen in
various manners according to the nature of the experimental situations which
it is desired to represent. Thus in the case of the Bush-Mosteller model for two
responses, the operator is given the form:
rz(") = ai, y,(n - 1) + atj