A Restricted Bayes Approach to Joint Detection and Estimation Under Prior Uncertainty

The problem of joint detection and estimation under prior uncertainty is considered within the framework of restricted Bayes theory, which covers the classical Bayes and minimax frameworks as special cases. A linear combination of the average estimation risk with respect to some guessed prior probability and the maximum conditional estimation risk is minimized for generic convex estimation cost functions subject to a constraint on a linear combination of the average detection error probability with respect to the assumed prior and the maximum conditional detection error probability. Unknown random parameters common to different hypotheses as well as parameters unique to each hypothesis are assumed. The jointly optimal estimators and detector that minimize the resulting estimation performance metric under the detection error constraint are derived. With the proposed framework, it is possible to strike any desired balance between a fully minimax framework (where the worst case performance metrics appear in both the objective and the constraint functions of the optimization problem) and the standard Bayesian setting where exact knowledge of the prior distribution is assumed to be available. Unlike the previous studies in the literature, which deal with special cases of the problem studied in this work, the solution to the most general problem indicates that the optimal estimators are coupled with the optimal detector through a least favorable prior.