Bayesian bounds for parameter estimation and nonlinear filtering /tracking

Bayesian estimation plays a central role in many signal processing problems encountered in radar, sonar, communications, seismology, and medical diagnosis. There are often highly nonlinear problems for which analytic evaluation of the exact performance is intractable. A widely used technique is to find bounds on the performance of any estimator and compare the performance of various estimators to these bounds.

This book provides a comprehensive overview of the state of the art in Bayesian Bounds. It addresses two related problems: the estimation of multiple parameters based on noisy measurements and the estimation of random processes, either continuous or discrete, based on noisy measurements.

An extensive introductory chapter provides an overview of Bayesian estimation and the interrelationship and applicability of the various Bayesian Bounds for both static parameters and random processes. It provides the context for the collection of papers that are included.

This book will serve as a comprehensive reference for engineers and statisticians interested in both theory and application. It is also suitable as a text for a graduate seminar or as a supplementary reference for an estimation theory course.

Author Notes

Harry L. Van Trees , ScD, is University Professor Emeritus at George Mason University. His career spans a variety of academic, government, and industry positions. He spent fourteen years as professor of electrical engineering at Massachusetts Institute of Technology. During this period he developed the three-volume series, Detection, Estimation, and Modulation Theory, which are the classical texts in the area and still widely used throughout the world. At George Mason University, he completed Part IV of the DEMT series, Optimum Array Processing. He served as chief scientist of the U.S. Air Force, principal deputy assistant secretary of defense for C3I, president of M/A-Com-Linkabit Government Systems, and founding director of the C3I Center at George Mason University. He is currently engaged in academic research at George Mason University, and is a consultant to several companies and government agencies.

Kristine L. Bell , PhD, is Associate Professor in the Department of Statistics and C3I Center at George Mason University. In her doctoral thesis, she developed a significant generalization of the Ziv-Zakai bound that expanded its range of applicability. Her current research interests are in the areas of robust, adaptive signal processing techniques and performance bounds for source location and tracking with applications in radar, sonar, aeroacoustics, and satellite communications.

Preface

IntroductionHarry L. Van Trees and Kristine L. Bell

1 Bayesian Estimation: Static Parameters

1.1 Maximum Likelihood and Maximum a Posteriori Estimation

1.1.1 Nonrandom Parameters

1.1.2 Random Parameters

1.1.3 Hybrid Parameters

1.1.4 Examples

1.2 Covariance Inequality Bounds

1.2.1 Covariance Inequality

1.2.2 Bayesian Bounds

1.2.3 Scalar Parameters

1.2.3.1 Bayesian Cram?r-Rao Bound

1.2.3.2 Weighted Bayesian Cram?r-Rao Bound

1.2.3.3 Bayesian Bhattacharyya Bound

1.2.3.4 Bobrovsky-Zakai Bound

1.2.3.5 Weiss-Weinstein Bound

1.2.4 Vector Parameters

1.2.4.1 Bayesian Cram?r-Rao Bound

1.2.4.2 Weighted Bayesian CRB

1.2.4.3 Bayesian Bhattacharyya Bound

1.2.4.4 Bobrovsky-Zakai Bound

1.2.4.5 Weiss-Weinstein Bound

1.2.5 Combined Bayesian Bounds

1.2.6 Nuisance Parameters

1.2.6.1 Nonrandom Unwanted Parameters

1.2.6.2 Random Parameters

1.2.7 Hybrid Parameters

1.2.8 Functions of the Parameter Vector

1.2.8.1 Scalar Parameters

1.2.8.2 Vector Parameters

1.2.9 Summary: Covariance Inequality Bounds

1.3 Ziv-Zakai Bounds

1.3.1 Scalar Parameters

1.3.2 Equally Likely Hypotheses

1.3.3 Vector Parameters

1.4 Method of Interval Estimation

1.5 Summary

2 Bayesian Estimation: Random Processes

2.1 Continuous-Time Processes and Continuous-Time Observations

2.1.1 Nonlinear Models

2.1.1.1 Linear AWGN Process and Observations

2.1.1.2 Linear AWGN Process, Nonlinear AWGN Observations

2.1.1.3 Nonlinear AWGN Process and Observations

2.1.1.4 Nonlinear Process and Observations

2.1.2 Bayesian Cram?r-Rao Bounds: Continuous-Time

2.2 Continuous-Time Processes and Discrete-Time Observations

2.2.1 Extended Kalman Filter

2.2.2 Bayesian Cram?r-Rao Bound

2.2.3 Discretizing the Continuous-Time State Equation

2.3 Discrete-Time Processes and Discrete-Time Observations

2.3.1 Linear AWGN Process and Observations

2.3.2 General Nonlinear Model

2.3.2.1 MMSE and MAP Estimation

2.3.2.2 Extended Kalman Filter

2.3.3 Recursive Bayesian Cram?r-Rao Bounds

2.4 Global Recursive Bayesian Bounds

2.5 Summary

3 Outline of the Book

Part I Bayesian Cram?r-Rao Bounds

1.1 Excerpts from Part I of Detection, Estimation, and Modulation Theory, pp. 66-86, Wiley, New York, 1968 (reprinted Wiley 2001)H. L. Van Trees

1.2 " A generalization of the Fr?chet-Cram?r inequality in the case of Bayes estimation," Bulletin of the American Mathematical Society, vol. 63, no. 142, 1957M. P. Shutzenberger

Part II Global Bayesian Bounds

2.1 "Some classes of global Cram?r-Rao bounds," Ann. Stat., vol. 15, pp. 1421-1438, 1987B. Z. Bobrovsky and E. Mayer-Wolf and M. Zakai

2.2 Excerpts from Part I of Detection, Estimation, and Modulation Theory, pp. 273-286, Wiley, New York, 1968 (reprinted 2001)H. L. Van Trees

2.3 "Single-tone parameter estimation from discrete-time observations," IEEE Trans. Inform. Theory, vol. IT-20, no. 5, pp. 591-598, September 1974D. Rife and R. Boorstyn

2.4 "Barankin bounds on parameter estimation," IEEE Trans. Info. Theory, vol. IT-17, no. 6, pp. 669-676, November 1971R. J. McAulay and E. M. Hostetter

2.5 "A modified Cram?r-Rao bound and its applications, IEEE Trans. Info. Theory, vol. 24, no. 3, pp. 398-400, May 1978R. Miller and C. Chang

2.6 "A lower bound on the mean-square error in random parameter estimation," IEEE. Trans. Info. Theory, vol. 31, no. 5, pp. 680-682A. Weiss and E. Weinstein

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Author Notes

Table of Contents