Bayesian sequential joint detection and estimation

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.2. The hypothesis H i , i = 0, 1, as well as the random parameter Θ i do not change during the observation period of X N .3. A sufficient static t n (x n ) in a state space (E t , E t ) exists such thatThe sufficient statistic has a transition kernel of the formand some initial statistic t 0 .4. The two hypotheses are separable, i.e.,5. The second order moment of the random parameter Θ exists and is finite, i.e., E[Θ 2 ] <∞ .Note that this implies, that the conditional second order moment E[Θ 2 | A] exist and is finite for all events A with non-zero probability.

An Asymptotically Pointwise Optimal Procedure For Sequential Joint Detection And Estimation

“…The problem of sequential joint signal detection and signal-to-noise ratio estimation was addressed in [15]. In [16], we developed a Bayesian framework for sequential joint detection and estimation for the binary hypothesis case. Contrary to the work by Yılmaz et al [11][12][13], the framework proposed in [16] comes with an approach to choose the coefficients, which parametrize the optimal policy, such that constraints on the detection and estimation errors are fulfilled and the resulting scheme uses on average as few samples as possible.…”

“…In [16], we developed a Bayesian framework for sequential joint detection and estimation for the binary hypothesis case. Contrary to the work by Yılmaz et al [11][12][13], the framework proposed in [16] comes with an approach to choose the coefficients, which parametrize the optimal policy, such that constraints on the detection and estimation errors are fulfilled and the resulting scheme uses on average as few samples as possible. The framework in [16] was then applied to sequential joint signal detection and signal-to-noise ratio estimation [17].…”

“…Contrary to the work by Yılmaz et al [11][12][13], the framework proposed in [16] comes with an approach to choose the coefficients, which parametrize the optimal policy, such that constraints on the detection and estimation errors are fulfilled and the resulting scheme uses on average as few samples as possible. The framework in [16] was then applied to sequential joint signal detection and signal-to-noise ratio estimation [17]. An extension to the multiple hypotheses scenario and its application to joint symbol decoding and noise power estimation is presented in [18,19].…”

“…The framework in [16] was then applied to sequential joint signal detection and signal-to-noise ratio estimation [17]. An extension to the multiple hypotheses scenario and its application to joint symbol decoding and noise power estimation is presented in [18,19]. In [20,21], sequential joint detection and estimation in distributed sensor networks is investigated.…”

“…Based on the results in [22], we proof that this policy is indeed APO for the sequential joint detection and estimation problem. Although the presented policy is not strictly optimal, it overcomes the drawback of the design approach presented in [16], namely, that the state space has to be discretized in order to design the test. This clearly becomes infeasible for higher dimensional spaces.…”