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This paper presents the results of a study of adaptive communication systems in which the "learning" is based on Bayes' Theorem. The learning is then used to guide adaptation whose purpose is to optimize system performance. Performance is evaluated by a measure based on decision theory. The communication system model used is a binary symmetric channel where "r" denotes the probability of correct transmission, and it is assumed that only the decoding is adaptable. Assuming a time invariant unknown r, it is shown that the true value of r will be learned in the limit, if, and only if, there is some redundancy in the message. It is also shown that the proposed adaptive system will converge to the best possible system if the true value of r is learned. The speed of convergence is measured by the probability that the best possible decoding system is selected after n messages have been received. This measure is calculated for a sample adaptive system in order to show the effect of redundancy in the transmitted message, the number of messages received, and the true value of r.