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

define more complex probabilistic contrasts that are possible when multiple cues
occur and can be affected by the context in which they are presented.
Empirically, the probability matching of perceptrons, illustrated in Figure 4-6,
suggests that their behaviour can represent P. When a cue is presented, activity
is equal to the probability that the cue signals reinforcement—that is, P(O|C). This
implies that the difference between a perceptron’s activity when a cue is presented
and its activity when a cue is absent must be equal to P. Let us now turn to a computational
analysis to prove this claim.
What is the formal relationship between formal contingency theories and
theories of associative learning (Shanks, 2007)? Researchers have compared the
predictions of an influential account of associative learning, the Rescorla-Wagner
model (Rescorla & Wagner, 1972), to formal theories of contingency (Chapman &
Robbins, 1990; Cheng, 1997; Cheng & Holyoak, 1995). It has been shown that while
in some instances the Rescorla-Wagner model predicts the conditional contrasts
defined by a formal contingency theory, in other situations it fails to generate these
predictions (Cheng, 1997).
Comparisons between contingency learning and Rescorla-Wagner learning
typically involve determining equilibria of the Rescorla-Wagner model. An equilibrium
of the Rescorla-Wagner model is a set of associative strengths defined by the
model, at the point where the asymptote of changes in error defined by Rescorla-
Wagner learning approaches zero (Danks, 2003). In the simple case described earlier,
involving a single cue and a single outcome, the Rescorla-Wagner model is identical
to contingency theory. This is because at equilibrium, the associative strength
between cue and outcome is exactly equal to P (Chapman & Robbins, 1990).
There is also an established formal relationship between the Rescorla-Wagner
model and the delta rule learning of a perceptron (Dawson, 2008; Gluck & Bower,
1988; Sutton & Barto, 1981). Thus by examining the equilibrium state of a perceptron
facing a simple contingency problem, we can formally relate this kind of network
to contingency theory and arrive at a formal understanding of what output
unit activity represents.
When a continuous activation function is used in a perceptron, calculus can be
used to determine the equilibrium of the perceptron. Let us do so for a single cue
situation in which some cue, C, when presented, is rewarded a frequency of a times,
and is not rewarded a frequency of b times. Similarly, when the cue is not presented,
the perceptron is rewarded a frequency of c times and is not rewarded a frequency
of d times. Note that to reward a perceptron is to train it to generate a desired
response of 1, and that to not reward a perceptron is to train it to generate a desired
response of 0, because the desired response indicates the presence or absence of the
unconditioned stimulus (Dawson, 2008).
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