Probability of error of some adaptive pattern-recognition machines

A simple taught pattern-recognition machine for detecting an unknown, fixed, randomly occurring pattern is derived using a Bayes´ approach, and its probability of error is analyzed. It is shown that with probability one, the machine converges to the optimal detector (a matched filter) for the unknown pattern, that the asymptotic decision function statistics are Gaussian, and that for an important class of problems, the central-limit theorem can be invoked to calculate the approximate probability of error at any stage of convergence. An untaught adaptive pattern-recognition machine may be made from the taught machine by using its own output instead of a teacher, and the asymptotic probability of error of this device is derived. It is shown that it does not converge to a matched filter for the unknown pattern, but that in any practical case it performs almost as well. Finally, the results of an experimental simulation of both machines are presented as curves of the relative frequency of error vs. time, and are compared with the values calculated by theory.