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A collection of the important papers dealing with the theory and application of Bayesian bounds. This book is suitable to both engineers and statisticians whether they are practitioners or theorists.
List of contents
Preface. Introduction ( Harry 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 Cramér Rao Bound. 1.2.3.2 Weighted Bayesian Cramér 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 Cramér 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 Cramér Rao Bounds: Continuous Time. 2.2 Continuous Time Processes and Discrete Time Observations. 2.2.1 Extended Kalman Filter. 2.2.2 Bayesian Cramér 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 Cramér Rao Bounds. 2.4 Global Recursive Bayesian Bounds. 2.5 Summary. 3 Outline of the Book. Part I Bayesian Cramér Rao Bounds. 1.1 H. L. Van Trees, Excerpts from Part I of Detection, Estimation, and Modulation Theory, pp. 66 86, Wiley, New York, 1968 (reprinted Wiley 2001). 1.2 M. P. Shutzenberger," A generalization of the Fréchet Cramér inequality in the case of Bayes estimation," Bulletin of the American Mathematical Society, vol. 63, no. 142, 1957. Part II Global Bayesian Bounds. 2.1 B. Z. Bobrovsky, E. Mayer Wolf, and M. Zakai, "Some classes of global Cramér Rao bounds," Ann. Stat., vol. 15, pp. 1421 1438, 1987. 2.2 H. L. Van Trees, Excerpts from Part I of Detection, Estimation, and Modulation Theory, pp. 273 286, Wiley, New York, 1968 (reprinted 2001). 2.3 D. Rife and R. Boorstyn, "Single tone parameter estimation from discrete time observations," IEEE Trans. Inform. Theory, vol. IT 20, no. 5, pp. 591 598, September 1974. 2.4 R. J. McAulay and E. M. Hostetter, "Barankin bounds on parameter estimation," IEEE Trans. Info. Theory, vol. IT 17, no. 6, pp. 669 676, November 1971. 2.5 R. Miller and C. Chang, "A modified Cramér Rao bound and its applications, IEEE Trans. Info. Theory, vol. 24, no. 3, pp. 398 400, May 1978. 2.6 A. Weiss and E. Weinstein, "A lower bound on the mean square error in random parameter estimation," IEEE. Trans. Info. Theory, vol. 31, no. 5, pp. 680 682, September 1985. 2.7 E. Weinstein and A. J. Weiss, "Lower bounds on the mean square estimation error," Proceedings of the IEEE, vol. 73, no. 9, pp. 1433 1434, September 1985. 2.8 E. Weinstein and A. J. Weiss, "A general class of lower bounds in parameter estimation," IEEE Trans. Info. Theory, vol. 34, no. 2, pp. 338 342, March 1988. 2.9 J. S. Abel, "A bound on mean square estimate error," IEEE. Trans. Info. Theory, vol. 39, no. 5, pp. 1675 1680, September 1993. 2.10 A. Renaux, P. Forster, P. Larzabal, and C. Richmond, "The Bayesian Abel bound on the mean square error," ICASSP 2006, vol. 3, pp. III 9 12, Toulouse,
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"This book will be useful as a text in an advanced seminar course on Bayesian estimation, and also as a reference for users." ( Computing Reviews May 2008)
