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According to the model discussed in this paper, a pattern recognizer is said to consist of two parts: a receptor, which generates a set of measurements of the physical sample to be recognized, and a categorizer, which assigns each set of measurements to one of a finite number of categories. The rule of operation of the categorizer is called the ``recognition function.'' The optimization of the recognition function is discussed, and the form of the optimal function is derived. In practice, a prohibitively large sample is required to provide a basis for estimating the optimal recognition function. If, however, certain assumptions about the probability distributions of the measurements are warranted, recognition functions that are asymptotically optimal may be obtained readily. A small numerical example, involving the recognition of the hand-printed characters A, B, and C is solved by means of the techiques described. The recognition accuracy is found to be 95 per cent.