This paper deals with the optimal use of a sequence of prior observations in order to recognize patterns. The class of pattern recognition problems which we treat is completely general in that the results may be applied to patterns of a visual, aural or electromagnetic origin. We show that the use of prior observations for pattern recognition may be described as a process of learning the statistical characteristics of the patterns involved. The mathematical description of the learning process is analyzed in order to obtain optimal (i.e., minimum probability of error) solutions to various types of pattern recognition problems. These solutions to the mathematical problem are then interpreted in terms of the systems implementation they imply. The optimum systems are shown to consist of banks of generalized correlators. Under assumption of independent measurements, the optimum systems reduce to a class of systems which function by reinforcing desired responses.

An optimum adaptive system is obtained for the identification of pattern samples which are the sum of a fixed unknown signal, determined by the pattern of the sample, plus Gaussian noise. The system learns the unknown signals from a set of pattern samples, called learning samples, which have been identified with absolute certainty. The adaptive system is optimum in the sense that it computes the a posteriori probability of each pattern, given the sample to be recognized and the learning samples. The rate at which the probability of misrecognition of the learning system approaches the probability of misrecognition of the a posteriori probability computing system with a priori knowledge of the fixed signals is derived, for binary recognition, as a function of the number of learning samples.