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This paper contains a theoretical analysis of the data association problem of a common type of surveillance system. By a method of inverse probability, the optimal data processor is obtained which permits maximum likelihood estimates to be made of the true data-surveillance object association. The maximum likelihood estimator is given in a form that lends itself to sequential computations performed in real time as the data arrives. Examples of the use of this estimator make clear the precise mathematical meaning of such terms as tentative, confirmed, and established data tracks, and the concept of search areas. The analytical technique is of general use in a variety of surveillance situations. Computer implementations are possible.