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

A second condition of the perceptron at equilibrium is that o ~c
is a value that causes
the derivative above to be equal to 0. As before, we can set the derivative to 0 and
solve for the value of o ~c
. This time the result is c/(c + d), which in a traditional contingency
table is equal to the conditional probability P(O|~C):
The main implication of the above equations is that they show that perceptron
activity is literally a conditional probability. This provides a computational proof for
the empirical hypothesis about perceptron activity that was generated from examining
Figure 4-6.
A second implication of the proof is that when faced with the same contingency
problem, a perceptron’s equilibrium is not the same as that for the Rescorla-Wagner
model. At equilibrium, the associative strength for the cue C that is determined by
Rescorla-Wagner training is literally P (Chapman & Robbins, 1990). This is not the
case for the perceptron. For the perceptron, P must be computed by taking the
difference between its output when the cue is present and its output when the cue
is absent. That is, P is not directly represented as a connection weight, but instead
is the difference between perceptron behaviours under different cue situations—
that is, the difference between the conditional probability output by the perceptron
when a cue is present and the conditional probability output by the perceptron
when the cue is absent.
Importantly, even though the perceptron and the Rescorla-Wagner model
achieve different equilibria for the same problem, it is clear that both are sensitive to
contingency when it is formally defined as P. Differences between the two reflect
an issue that was raised in Chapter 2, that there exist many different possible algorithms
for computing the same function. Key differences between the perceptron
and the Rescorla-Wagner model—in particular, the fact that the former performs a
nonlinear transformation on internal signals, while the latter does not—cause them
to adopt very different structures, as indicated by different equilibria. Nonetheless,
these very different systems are equally sensitive to exactly the same contingency.
This last observation has implications for the debate between contingency theory
and associative learning (Cheng, 1997; Cheng & Holyoak, 1995; Shanks, 2007). In
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