Constrained testing and estimation for Bayesian joint decision and estimation

This paper studies the problem of simultaneous deciding on hypotheses and estimating a random parameter. We propose a joint decision and estimation (JDE) formulation, which amounts to minimizing a risk related to both decision and estimation while decision performance is also constrained within a tolerable level. The risk used in this paper is a weighted sum of estimation costs conditioned on correct decisions times the correct decision probabilities. Such a risk allows us to focus on the joint (decision and estimation) performance pertinent to correct decisions only. The JDE solution includes handling two subproblems - estimation and decision. For the estimation subproblem, we derive the optimal Bayes estimator. For the decision subproblem, we provide the optimal solution via a numerical algorithm of a two-dimensional search. To reduce the complexity, we also provide a suboptimal solution when the decision rule is confined to the likelihood ratio tests. The suboptimal solution is implemented with a one-dimensional search in a finite range. Numerical results are provided to validate our methods.