  A previous paper [2] showed how to generate a linear discriminant network
(LDN) that computes likely faults for a noisy fault detection problem by using
a modification of the perceptron learning algorithm called the pocket
algorithm. Here we compare the performance of this connectionist model with
performance of the optimal Bayesian decision rule for the example that was
previously described. We find that for this particular problem the
connectionist model performs about 97% as well as the optimal Bayesian
procedure. We then define a more general class of noisy single-pattern boolean
(NSB) fault detection problems where each fault corresponds to a single
:pattern of boolean instrument readings and instruments are independently
noisy. This is equivalent to specifying that instrument readings are
probabilistic but conditionally independent given any particular fault. We
prove:
  1. The optimal Bayesian decision rule for every NSB fault detection problem
is representable by an LDN containing no intermediate nodes. (This slightly
extends a result first published by Minsky & Selfridge.) 2. Given an NSB fault
detection problem, then with arbitrarily high probability after sufficient
iterations the pocket algorithm will generate an LDN that computes an optimal
Bayesian decision rule for that problem. In practice we find that a reasonable
number of iterations of the pocket algorithm produces a network with good, but
not optimal, performance.
