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

Assume that when the cue is present, the logistic activation function computes
an activation value that we designate as o c
, and that when the cue is absent it returns
the activation value designated as o ~c
. We can now define the total error of responding
for the perceptron, that is, its total error for the (a + b + c + d) number of patterns
that represent a single epoch, in which each instance of the contingency problem is
presented once. For instance, on a trial in which C is presented and the perceptron
is reinforced, the perceptron’s error for that trial is the squared difference between
the reward, 1, and o c.
As there are a of these trials, the total contribution of this type
of trial to overall error is a(1 – o c
) 2 . Applying this logic to the other three pairings of
cue and outcome, total error E can be defined as follows:
For a perceptron to be at equilibrium, it must have reached a state in which total
error has been optimized, so that the error can no longer be decreased by using the
delta rule to alter the perceptron’s weight. To determine the equilibrium of the perceptron
for the single cue contingency problem, we begin by taking the derivative
of the error equation with respect to the activity of the perceptron when the cue is
present, o c
:
One condition of the perceptron at equilibrium is that o c
is a value that causes this
derivative to be equal to 0. The equation below sets the derivative to 0 and solves for
o c
. The result is a/(a + b), which is equal to the conditional probability P(O|C) if the
single cue experiment is represented with a traditional contingency table:
Similarly, we can take the derivative of the error equation with respect to the activity
of the perceptron when the cue is not present, o~c:
156 Chapter 4