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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.