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We consider the problem of jointly testing multiple hypotheses and estimatinga random parameter of the underlying distribution. This problem is investigatedin a sequential setup under mild assumptions on the underlying random process.The optimal method minimizes the expected number of samples while ensuring thatthe average detection/estimation errors do not exceed a certain level. Afterconverting the constrained problem to an unconstrained one, we characterize thegeneral solution by a non-linear Bellman equation, which is parametrized by aset of cost coefficients. A strong connection between the derivatives of thecost function with respect to the coefficients and the detection/estimationerrors of the sequential procedure is derived. Based on this fundamentalproperty, we further show that for suitably chosen cost coefficients thesolutions of the constrained and the unconstrained problem coincide. We presenttwo approaches to finding the optimal coefficients. For the first approach, thefinal optimization problem is converted into a linear program, whereas thesecond approach solves it with a projected gradient ascent. To illustrate thetheoretical results, we consider two problems for which the optimal schemes aredesigned numerically. Using Monte Carlo simulations, it is validated that thenumerical results agree with the theory.

Bayesian Sequential Joint Detection and Estimation under Multiple Hypotheses