Optimal joint detection and estimation in linear models

The problem of optimal joint detection and estimation in linear models with Gaussian noise is studied. A simple closed-form expression for the joint posterior distribution of the (multiple) hypotheses and the states is derived. The expression crystalizes the dependence of the optimal detector on the state estimates. The joint posterior distribution characterizes the beliefs ("soft information") about the hypotheses and the values of the states. Furthermore, it is a sufficient statistic for jointly detecting multiple hypotheses and estimating the states. The developed expressions give us a unified framework for joint detection and estimation under all performance criteria.