Mind, Body, World- Foundations of Cognitive Science, 2013a

Perceptrons, too, can match probabilities (Dawson et al., 2009). Dawson et al.
used four different cues, or discriminative stimuli (DSs), but did not “reward” them
100 percent of the time. Instead, they rewarded one DS 20 percent of the time,
another 40 percent, a third 60 percent, and a fourth 80 percent. After 300 epochs,
where each epoch involved presenting each cue alone 10 different times in random
order, these contingencies were inverted (i.e., subtracted from 100). The dependent
measure was perceptron activity when a cue was presented; the activation function
employed was the logistic. Some results of this experiment are presented in Figure
4-6. It shows that after a small number of epochs, the output unit activity becomes
equal to the probability that a presented cue was rewarded. It also shows that perceptron
responses quickly readjust when contingencies are suddenly modified, as
shown by the change in Figure 4-6 around epoch 300. In short, perceptrons are
capable of probability matching.
Mean Network Activation
Epoch of Training
Figure 4-6. Probability matching by perceptrons. Each line shows the perceptron
activation when a different cue (or discriminative stimulus, DS) is presented.
Activity levels quickly become equal to the probability that each cue was
reinforced (Dawson et al., 2009).
That perceptrons match probabilities relates them to contingency theory. Formal
statements of this theory formalize contingency as a contrast between conditional
probabilities (Allan, 1980; Cheng, 1997; Cheng & Holyoak, 1995; Cheng & Novick,
1990, 1992; Rescorla, 1967, 1968).
For instance, consider the simple situation in which a cue can either be presented,
C, or not, ~C. Associated with either of these states is an outcome (e.g., a
reward) that can either occur, O, or not, ~O. In this simple situation, involving a
single cue and a single outcome, the contingency between the cue and the outcome
is formally defined as the difference in conditional probabilities, P, where P =
P(O|C) – P(O|~C) (Allan, 1980). More sophisticated models, such as the probabilistic
contrast model (e.g., Cheng & Novick, 1990) or the power PC theory (Cheng, 1997),
154 Chapter 4