Bayesian Sequential Joint Detection and Estimation

TL;DR

This paper develops a framework for joint detection and estimation in a sequential setting, linking the cost function derivatives to errors, and solving constrained problems efficiently with linear programming, validated through examples.

Contribution

It introduces a novel connection between cost derivatives and errors, and provides an efficient method to solve constrained joint detection and estimation problems.

Findings

Optimal schemes designed numerically for example problems

Solution of constrained problems via linear programming

Performance validated through Monte Carlo simulations

Abstract

Joint detection and estimation refers to deciding between two or more hypotheses and, depending on the test outcome, simultaneously estimating the unknown parameters of the underlying distribution. This problem is investigated in a sequential framework under mild assumptions on the underlying random process. We formulate an unconstrained sequential decision problem, whose cost function is the weighted sum of the expected run-length and the detection/estimation errors. Then, a strong connection between the derivatives of the cost function with respect to the weights, which can be interpreted as Lagrange multipliers, and the detection/estimation errors of the underlying scheme is shown. This property is used to characterize the solution of a closely related sequential decision problem, whose objective function is the expected run-length under constraints on the average detection/estimation errors. We show that the solution of the constrained problem coincides with the solution of the unconstrained problem with suitably chosen weights. These weights are characterized as the solution of a linear program, which can be solved using efficient off-the-shelf solvers. The theoretical results are illustrated with two example problems, for which optimal sequential schemes are designed numerically and whose performance is validated via Monte Carlo simulations.