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570 Raymond Boudon
(sign that Az is the correct response). In the experiment we are considering, the
frequency n with which one of the responses, for example A,, is reinforced,
remains independent of the subject's responses. The formalization which
Estes and Burke produce for this learning process is as follows:
I. In any given experiment, the subject is confronted with a set S, which breaks
down a stimulus into a ked number s of elements.
2. At each stage, each element is connected either with response Ax, or with
response A,.
3. At each stage, the subject chooses at random a proportion 8 of the elements.
4. If among the 8 elements picked at random, i elements are connected with AI
and j with A,, then the subject chooses AI with the probability i/(i+j) = i/s.
5. If the investigator chooses E,, those of the elements in the sample which were
connected with Az become connected with AI and vice versa. The elements
outside the sample do not change their connexions.
This is obviously an abstract theoretical scheme. In some experiments, simple
representations of the elements of the stimulus have been constructed. But it is
not necessary to produce physical interpretations of these elements : they can
be treated as real, but not observable. Their function is simply to introduce the
parameter 8 which can be taken as a parameter measuring the effect of the
reinforcement. According as the value of 8 is greater or smaller, the conditioning
will change at a faster or slower rate. The advantage of the symbolic representation
of 0 under the form of a set of elements is that it enables learning to be represented
as a stochastic process, where the elements play the part of the balls
in an urn model. The learning curve is deduced from the process just described,
starting from the assumption that at each stage of the experiment the subject
can be deemed to be in one of the following states: state o when none of the s
elements is connected with A,; state I when I element is connected with A,, etc.
Making the assumption that s = 2, the probability of passing from state
o to state I then equals:
(Probability of the experimenter reinforcing response A, by E,) x (Probability
of one element being connected to A, and the others not).
Whence :
pox = n x 2e(1- e)
The whole range of transition probabilities, Poo, POI, PO2, ... Pz2, can thus be
calculated, forming the matrix P.
Next we consider the probability of response A, at the nth repetition:
rz(") = (o)P0(")
+ (l/s)PI(") + ... + (s/s)P8(")
where the quantities (o), (I/s), ... (s/s) are the probabilities of subjects giving a
positive response to A, when in states 0, I, ... s. From this is deduced:
It(") = Z-(I- 8)n[n-r,(0)]
We thus see that the probability of response A, at the nth repetition is a func-
tion of rzto), of 8 and IC. 8 is a parameter which measures the rapidity of learn-
ing; n is the probability of response A, at the conclusion of the learning process
(i.e. when n + 00, rz(n) -+ n).