TY - JOUR
T1 - A fresh look at the Bayesian bounds of the Weiss-Weinstein family
AU - Renaux, Alexandre
AU - Forster, Philippe
AU - Larzabal, Pascal
AU - Richmond, Christ D.
AU - Nehorai, Arye
N1 - Funding Information:
Manuscript received April 24, 2007; revised April 18, 2008 First published June 13, 2008; current version published October 15, 2008. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Philippe Loubaton. The work of A. Renaux and A. Nehorai was supported in part by the Department of Defense under the Air Force Office of Scientific Research MURI Grant FA9550-05-1-0443, AFOSR Grant FA9550-05-1-0018, and by the National Science Foundation by Grants CCR-0330342 and CCF-0630734. The material in this paper was presented in part at the IEEE Workshop on Statistical Signal Processing, Bordeaux, France, July 2005 and at the IEEE International Conference on Acoustic, Speech and Signal Processing, Toulouse, France, May 2006.
PY - 2008
Y1 - 2008
N2 - Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss-Weinstein family. Among this family, we have Bayesian Cramér-Rao bound, the Bobrovsky-MayerWolf-Zakaï bound, the Bayesian Bhattacharyya bound, the Bobrovsky-Zakaï bound, the Reuven-Messer bound, and the Weiss-Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer-Wolf, and Zakaï. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the Reuven-Messer bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakaï bound, and the Bayesian Cramér-Rao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem.
AB - Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss-Weinstein family. Among this family, we have Bayesian Cramér-Rao bound, the Bobrovsky-MayerWolf-Zakaï bound, the Bayesian Bhattacharyya bound, the Bobrovsky-Zakaï bound, the Reuven-Messer bound, and the Weiss-Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer-Wolf, and Zakaï. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the Reuven-Messer bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakaï bound, and the Bayesian Cramér-Rao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem.
KW - Bayesian bounds on the MSE
KW - Weiss-Weinstein family
UR - http://www.scopus.com/inward/record.url?scp=54949113694&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=54949113694&partnerID=8YFLogxK
U2 - 10.1109/TSP.2008.927075
DO - 10.1109/TSP.2008.927075
M3 - Article
AN - SCOPUS:54949113694
SN - 1053-587X
VL - 56
SP - 5334
EP - 5352
JO - IRE Transactions on Audio
JF - IRE Transactions on Audio
IS - 11
ER -