Bayesian sequential joint detection and estimation under multiple hypotheses

Abstract

We consider the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution. This problem is investigated in a sequential setup under mild assumptions on the underlying random process. The optimal method minimizes the expected number of samples while ensuring that the average detection/estimation errors do not exceed a certain level. After converting the constrained problem to an unconstrained one, we characterize the general solution by a nonlinear Bellman equation, which is parameterized by a set of cost coefficients. A strong connection between the derivatives of the cost function with respect to the coefficients and the detection/estimation errors of the sequential procedure is derived. Based on this fundamental property, we further show that for suitably chosen cost coefficients the solutions of the constrained and the unconstrained problem coincide. We present two approaches to finding the optimal coefficients. For the first approach, the final optimization problem is converted into a linear program, whereas the second approach solves it with a projected gradient ascent. To illustrate the theoretical results, we consider two problems for which the optimal schemes are designed numerically. Using Monte Carlo simulations, it is validated that the numerical results agree with the theory.

Keywords:

• Joint detection and estimation
• multiple hypotheses
• sequential analysis
• stopping time

SUBJECT CLASSIFICATIONS:

• 62L10
• 62L12
• 62L15
• 90C05
• 93E10

ACKNOWLEDGMENTS

The authors thank the anonymous reviewers and the editors for their time and effort.

DISCLOSURE

The authors have no conflicts of interest to report.

Additional information

Funding

The work of Dominik Reinhard was supported by the German Research Foundation (DFG) under grant number 390542458. The work of Michael Fauß was supported by the German Research Foundation (DFG) under grant number 424522268.