A posteriori distributions and detection theory

Abstract

The notion of a posteriori probability, often used in hypothesis testing in connection with problems of optimum signal detection, is put on a firm basis. The number of hypotheses is countable, and the observation space ω is abstract so as to include the case where the observation is a realization of a continuous parameter random process. The a posteriori probability is defined without recourse to limiting arguments on “finite dimensional≓ conditional probabilities. The existence of the a posteriori probability is established, its a.e. uniqueness is studied, and it is then used to define other a posteriori quantities and to solve the decision problem of minimizing the error probability. In particular, a precise version of the loose assertion that “minimizing the error probability is equivalent to maximizing the a posteriori probability≓ is stated and proved. The results are then applied to the case where the observation is a sample path of a random process, devoting considerable attention to questions of convergence and of having satisfactory models for the observation space, the random process, and the observables. The deficiencies of a common function space type model are pointed out and ways of correcting these deficiencies are discussed. The use of time samples and Karhunen-Loève expansion coefficients as observables is investigated. The paper closes with an examination of non function space type models, and a demonstration that function space type models are, in a sense, natural models for detection problems.