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Library of Congress Cataloging-in-Publication Data:

Bayesian signal processing : classical, modern, and particle filtering methods / James V. Candy., p. cm.

Includes bibliographical references.

ISBN 978-0-470-18094-5 (cloth)

1. Signal processing—Mathematics. 2. Bayesian statistical decision theory. I. Title. TK5102.9.C3187 2008 621.382’2-dc22

2008032184

Peace, real peace, can only be found through the Lord, our Savior, Jesus Christ!

Preface

In the real world, systems designed to extract signals from noisy measurements are plagued by errors evolving from constraints of the sensors employed, to random disturbances and noise and probably, most common, the lack of precise knowledge of the underlying physical phenomenology generating the process in the first place! Methods capable of extracting the desired signal from hostile environments require approaches that capture all of the a priori information available and incorporate them into a processing scheme. This approach is typically model-based [1] employing mathematical representations of the component processes involved. However, the actual implementation providing the algorithm evolves from the realm of statistical signal processing using a Bayesian approach based on Bayes’ rule. Statistical signal processing is focused on the development of processors capable of extracting the desired information from noisy, uncertain measurement data. This is a text that develops the “Bayesian approach” to statistical signal processing for a variety of useful model sets. It features the next generation of processors which have recently been enabled with the advent of high speed/high throughput computers. The emphasis is on nonlinear/non-Gaussian problems, but classical techniques are included as special cases to enable the reader familiar with such methods to draw a parallel between the approaches. The common ground is the model sets. Here the state-space approach is emphasized because of its inherent applicability to a wide variety of problems both linear and nonlinear as well as time invariant and time-varying problems including what has become popularly termed “physics-based” models. This text brings the reader from the classical methods of model-based signal processing including Kalman filtering for linear, linearized and approximate nonlinear processors as well as the recently developed unscented or sigma-point filters to the next generation of processors that will clearly dominate the future of model-based signal processing for years to come. It presents a unique viewpoint of signal processing from the Bayesian perspective in contrast to the pure statistical approach found in many textbooks. Although designed primarily as a graduate textbook, it will prove very useful to the practicing signal processing professional or scientist, since a wide variety of applications are included to demonstrate the applicability of the Bayesian approach to real-world problems. The prerequisites for such a text is a melding of undergraduate work in linear algebra, random processes, linear systems, and digital signal processing as well as a minimal background in model-based signal processing illustrated in the recent text [1]. It is unique in the sense that few texts cover the breadth of its topics, whereas, the underlying theme of this text is the Bayesian approach that is uniformly developed and followed throughout in the algorithms, examples, applications and case studies. It is this theme coupled with the hierarchy of physics-based models developed that contribute to its uniqueness. This text has evolved from three previous texts, Candy [1-3] coupled with a wealth of practical applications to real-world Bayesian problems.

The Bayesian approach has existed in statistical physics for a long time and can be traced back to the 1940s with the evolution of the Manhattan project and the work of such prominent scientists as Ulam, von Neumann, Metropolis, Fermi, Feynman, and Teller. Here the idea of Monte Carlo (MC) techniques to solve complex integrals evolved [4]. Since its birth, Monte Carlo related methods have been the mainstay of many complex statistical computations. Many applications have evolved from this method in such areas as physics, biology, chemistry, computer science, economics/finance, material science, statistics and more recently in engineering. Thus, statisticians have known for a long time about these methods, but their practicalities have not really evolved as a working tool until the advent of high speed super computers around the 1980s. In signal processing it is hard to pinpoint the actual initial starting point but clearly the work of Handschin and Mayne in the late 1960s and early 1970s [5, 6] was the initial evolution of Monte Carlo techniques for signal processing and control. However from the real-time perspective, it is probably the development of the sequential Bayesian processor made practical by the work of Gordon, Salmond and Smith in 1993 [7] enabling the evolution and the explosion of the Bayesian sequential processor that is currently being researched today. To put this text in perspective we must discuss the current signal processing texts available on Bayesian processing. Since its evolution much has been published in the statistical literature on Bayesian techniques for statistical estimation; however, the earliest texts are probably those of Harvey [8], Kitigawa and Gersch [9] and West [10] which emphasize the Bayesian model-based approach incorporating dynamic linear or nonlinear models into the processing scheme for additive Gaussian noise sources leading to the classical approximate (Kalman) filtering solutions. These works extend those results to non-Gaussian problems using Monte Carlo techniques for eventual solution laying the foundation for works to follow. Statistical MC techniques were also available, but not as accessible to the signal processor due to statistical jargon and abstractness of the discussions. Many of these texts have evolved during the 1990s such as Gilks [11], Robert [12], Tanner [13], Tanizaki [14], with the more up-to-date expositions evolving in the late 1990s and currently such as Liu [4], Ruanaidh [15], Haykin [16], Doucet [17], Ristic [18] and Cappe [19]. Also during the last period a sequence of tutorials and special IEEE issues evolved exposing the MC methods to the signal processing community such as Godsill [20], Arulampalam [21], Djuric [22], Haykin [23] and Doucet [24], Candy [25], as well as a wealth of signal processing papers (see references for details). Perhaps the most complete textbook from the statistical researcher’sperspective is that of Cappe [19]. In this text much of the statistical MC sampling theory is developed along with all of the detailed mathematics—ideal for an evolving researcher. But what about the entry level person—the engineer, the experimentalist, and the practitioner? This is what is lacking in all of this literature. Questions like, how do the MC methods relate to the usual approximate Kalman methods? How does one incorporate models (model-based methods) into a Bayesian processor? How does one judge performance compared with classical methods? These are all basically pragmatic questions that the proposed text will answer in a lucid manner through coupling the theory to real-world examples and applications. Thus, the goal of this text is to provide a bridge for the practitioners with enough theory and applications to provide the basic background to comprehend the Bayesian framework and enable the application of these powerful techniques to real-world problem solving. Next, let us discuss the structure of the proposed text in more detail to understand its composition and approach.

We first introduce the basic ideas and motivate the need for such processing while showing that they clearly represent the next generation of processors. We discuss potential application areas and motivate the requirement for such a generalization. That is, we discuss how the simulation-based approach to Bayesian processor design provides a much needed capability, while well known in the statistical community, not very well known (until recently) in the signal processing community. After introducing the basic concepts in Chapter 1, we begin with the basic Bayesian processors in Chapter 2. We start with the Bayesian “batch” processor and establish its construction by developing the fundamental mathematics required. Next we discuss the well-known maximum likelihood (ML) and minimum (error) variance (MV) or equiv-alently minimum mean-squared error (MMSE) processors. We illustrate the similarity and differences between the schemes. Next we launch into sequential Bayesian processing schemes which forms the foundation of the text. By examining the “full” posterior distribution in both dynamic variables of interest as well as the full data set, we are able to construct the sequential Bayesian approach and focus on the usual filtered or filtering distribution case of highest interest demonstrating the fundamental prediction/update recursions inherent in the sequential Bayesian structure. Once establishing the general Bayesian sequential processor (BSP) the schemes that follow are detailed depending on the assumed distribution with a variety of model sets.

We briefly review simulation-based methods starting with sampling methods, progressing to Monte Carlo approaches leading to the basic iterative methods of sampling using the Metropolis, Metropolis-Hastings, Gibb’s and slice samplers. Since one of the major motivations of recursive or sequential Bayesian processing is to provide a real-time or pseudo real-time processor, we investigate the idea of importance sampling as well as sequential importance sampling techniques leading to the generic Bayesian sequential importance sampling algorithm. Here we show the solution can be applied, once the importance sampling distribution is defined.

In order to be useful, Bayesian processing techniques must be specified through a set of models that represent the underlying phenomenology driving the particular application. For example, in radar processing we must investigate the propagation models, tracking models, geometric models, and so forth. In Chapter 4, we develop the state-space approach to signal modeling which forms the basis of many applications such as speech, radar, sonar, acoustics, geophysics, communications, control, etc. Here we investigate continuous, sampled-data and discrete state-space signals and systems. We also discuss the underlying systems theory and extend the model-set to include the stochastic case with noise driving both process and measurements leading the well-known Gauss-Markov (GM) representation which forms the starting point for the classical Bayesian processors to follow. We also discuss the equivalence of the state-space model to a variety of time series (ARMA, AR, MA, etc.) representations as well as the common engineering model sets (transfer functions, all-pole, all-zero, pole-zero, etc.). This discussion clearly demonstrates why the state-space model with its inherent generality is capable of capturing the essence of a broad variety of signal processing representations. Finally, we extend these ideas to nonlinear statespace models leading to “approximate” Gauss-Markov representation evolving from nonlinear, perturbed and linearized systems.

In the next chapter, we develop classical Bayesian processors by first motivating the Bayesian approach to the state-space where the required conditional distributions use the embedded state-space representation. Starting with the linear, time-varying, state-space models, we show that the “optimum” classical Bayesian processor under multivariate Gaussian assumptions leads to minimum (error) variance (MV) or equiv-alently minimum mean-squared error (MMSE), which is the much heralded Kalman filter of control theory [1]. That is, simply substituting the underlying Gauss-Markov model into the required conditional distributions leads directly to the BSP or Kalman filter in this case. These results are then extended to the nonlinear state-space representation which are linearized using a known reference trajectory through perturbation theory and Taylor-series expansions. Starting with the linearized or approximate GM model of Chapter 4, we again calculate the required Bayesian sequential processor from the conditionals which lead to the “linearized” BSP (or linearized Kalman filter) algorithm. Once this processor is developed, it is shown that the “extended” Bayesian processor follows directly by linearizing about the most currently available estimate rather than the reference trajectory. The extended Bayesian processor (XBP) or equiv-alently extended Kalman filter (EKF) of nonlinear processing theory evolves quite naturally from the Bayesian perspective, again following the usual development by defining the required conditionals, making nonlinear approximations and developing the posterior distributions under multivariate Gaussian assumptions. Next, we briefly investigate an iterative version of the XBP processor, again from the Bayesian perspective which leads directly to the iterative version of the extended Bayesian processor (IX-BP) algorithm—an effective tool when nonlinear measurements dominate the uncertain measurements required.

Chapter 6 focuses on statistical linearization methods leading to the modern unscented Bayesian processor (UBP) or equivalently sigma-point Bayesian processor (SPBP). Here we show how statistical linearization techniques can be used to transform the underlying probability distribution using the sigma-point or unscented nonlinear transformation technique (linear regression) leading to the unscented Bayesian processor or equivalently the unscented Kalman filter (UKF). Besides developing the fundamental theory and algorithm, we demonstrate its performance on a variety of example problems. We also briefly discuss the Gaussian-Hermite quadrature (G-H) and Gaussian sum (G-S) techniques for completeness.

We reach the heart of the particle filtering methods in Chapter 7, where we discuss the Bayesian approach to the state-space. Here the ideas of Bayesian and modelbased processors are combined through the development of Bayesian state-space particle filters. Initially, it is shown how the state-space models of Chapter 4 are incorporated into the conditional probability distributions required to construct the sequential Bayesian processors through importance sampling constructs. After investigating a variety of importance proposal distributions, the basic set of state-space particle filters (SSPF) are developed and illustrated through a set of example problems and simulations. The techniques including the Bootstrap, auxiliary, regularized MCMC and linearized particle filters are developed and investigated when applied to the set of example problems used to evaluate algorithm performance.

In Chapter 8 the important joint Bayesian SSPF are investigated by first developing the joint filter popularly known as the parametrically adaptive processor [1]. Here both states and static as well as dynamic parameters are developed as solutions to this joint estimation problem. The performance of these processors are compared to classical and modern processors through example problems.

In Chapter 9 the hidden Markov models (HMM) are developed for event related problems (e.g., Poisson point processes). This chapter is important in order to place purely discrete processes into perspective. HMM evolve for any type of memoryless, counting processes and become important in financial applications, communications, biometrics, as well as radiation detection. Here we briefly develop the fundamental ideas and discuss them in depth to develop a set of techniques used by the practicioner while applying them to engineering problems of interest.

In the final chapter, we investigate a set of physics-based applications focusing on the Bayesian approach to solving real-world problems. By progressing through a step-by-step development of the processors, we see explicitly how to develop and analyze the performance of such Bayesian processors. We start with a practical laser alignment problem followed by a broadband estimation problem in ocean acoustics. Next the solid-state microelectromechanical (MEM) sensor problem for biothreat detection is investigated followed by a discrete radiation detection problem based on counting statistics. All of these methods invoke Bayesian techniques to solve the particular problems of interest enabling the practitioner the opportunity to track “real-world” Bayesian model-based solutions.

The place of such a text in the signal processing textbook community can best be explained by tracing the technical ingredients that comprise its contents. It can be argued that it evolves from the digital signal processing area primarily from those texts that deal with random or statistical signal processing or possibly more succinctly “signals contaminated with noise.” The texts by Kay [26-28], Therrien [29], Brown [30] all provide the basic background information in much more detail than this text, so there is little overlap at the detailed level with them.

This text, however, possesses enough theory for the graduate or advanced graduate student to develop a fundamental basis to go onto more rigorous texts like Jazwinski [31], Sage [32], Gelb [33], Anderson [34], Maybeck [35], Bozic [36], Kailath [37, 38], and more recently, Mendel [39], Grewel [40], Bar-Shalom [41] and Simon [42]. These texts are rigorous and tend to focus on Kalman filtering techniques ranging from continuous to discrete with a wealth of detail on all of their variations. The Bayesian approach discussed in this text certainly includes the state-space models as one of its model classes (probably the most versatile), but the emphasis is on various classes of models and how they may be used to solve a wide variety of signal processing problems. Some of the more recent texts about the same technical level, but again, with a different focus: are Widrow [43], Orfanidis [44], Sharf [45], Haykin [46], Hayes [47], Brown [30] and Stoica [48]. Again the focus of these texts is not the Bayesian approach but more on narrow set of specific models and the development of a variety of algorithms to estimate these sets. The system identification literature and texts therein also provide some overlap with this text, but again the approach is focused on estimating a model from noisy data sets and not really aimed at developing a Bayesian solution to a particular signal processing problem. The texts in this area are Ljung [49, 50], Goodwin [51], Norton [52] and Soderstrom [53].

The recent particle filtering texts of Ristic [18] and Cappe [19] are useful as references to accompany this text, especially if more details are required on the tracking problem and the fundamental theorems governing statistical properties and convergence proofs. That is, Ristic’s text provides a introduction that closely follows the 2002 tutorial paper by Arulampalam [21] but provides little of the foundational material necessary to comprehend this approach. It focuses primarily on the tracking problem. Cappe’s text is at a much more detailed technical level and is written for researcher’s in this area not specifically aimed at the practitioner’s viewpoint. The proposed text combines the foundational material, some theory along with the practice and application of PF to real-world applications and examples.

The approach we take is to introduce the basic idea of Bayesian signal processing and show where it fits in terms of signal processing. It is argued that BSP is a natural way to solve basic processing problems. The more a priori information we know about data and its evolution, the more information we can incorporate into the processor in the form of mathematical models to improve its overall performance. This is the theme and structure that echoes throughout the text. Current applications (e.g., structures, tracking, equalization, biomedical) and simple examples are incorporated to motivate the signal processor. Examples are discussed to motivate all of the models and prepare the reader for further developments in subsequent chapters. In each case the processor along with accompanying simulations are discussed and applied to various data sets demonstrating the applicability and power of the Bayesian approach. The proposed text is linked to the MATLAB (signal processing standard software) software package providing notes at the end of each chapter.

In summary, this Bayesian signal processing text will provide a much needed “down-to-earth” exposition of modern MC techniques. It is coupled with well-known signal processing model sets along with examples and problems that can be used to solve many real-world problems by practicing engineers and scientists alongwith entry level graduate students as well as advanced undergraduates and post-doctorates requiring a solid introduction to the “next generation” of model-based signal processing techniques.

JAMES V. CANDY

Danville, California

January 2009

REFERENCES TO THE PREFACE

1. J. Candy, Model-Based Signal Processing (Hoboken, NJ: John Wiley/IEEE Press, 2006).

2. J. Candy, Signal Processing: The Model-Based Approach (New York: McGraw-Hill, 1986).

3. J. Candy, Signal Processing: The Modern Approach (New York: McGraw-Hill, 1988).

4. J.Liu, Monte Carlo Strategies in Scientific Computing (New York: Springer-Verlag, 2001).

5. J. Handschin and D. Mayne, “Monte Carlo techniques to estimate the conditional expectation in multi-stage nonlinear filtering,” Intl. J. Control, 9, 547-559, 1969.

6. J. Handschin, “Monte Carlo techniques for prediction and filtering of nonlinear stochastic processes,” Automatica, 6, 555-563, 1970.

7. N. Gordon, D. Salmond and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proc.-F., 140, 107-113, 1993.

8. A. Harvey, Forcasting, Structural Time Series Models and the Kalman Filter (Cambridge, UK: Cambridge University Press, 1989).

9. G. Kitagawa and W. Gersch, Smoothness Priors Analysis of Time Series (New York: Springer-Verlag, 1996).

10. M. West and J. Harrison, Bayesian Forcasting and Dynamic Models, 2n Ed. (New York: Springer-Verlag, 1997).

11. W. Gilks, S. Richardson and D. Spiegelhalter, Markov Chain Monte Carlo in Practice (New York: Chapman & Hall/CRC Press, 1996).

12. C. Robert and G. Casella, Monte Carlo Statistical Methods (New York: Springer, 1999).

13. M. Tanner, Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, 2nd Ed. (New York: Springer-Verlag, 1993).

14. H.Tanizaki, “Nonlinear filters: Estimation and Applications.” Lecture Notes in Economics andMath. Sys. (New York: Springer, 1993).

15. J. Ruanaidh and W. Fitzgerald, Numerical Bayesian Methods Applied to Signal Processing (New York: Springer-Verlag, 1996).

16. S. Haykin, Kalman Filtering and Neural Networks (Hoboken, NJ: Wiley: 2001).

17. A. Doucet, N. de Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice, (New York: Springer-Verlag, 2001).

18. B. Ristic, S. Arulampalam and N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracking Applications (Boston: Artech House, 2004).

19. O. Cappe, E. Moulines and T. Ryden, Inference in Hidden Markov Models (New York: Springer-Verlag, 2005).

20. S. Godsill and P. Djuric, “Special Issue: Monte Carlo methods for statistical signal processing.” IEEE Trans. Signal Proc., 50, 173-499, 2002.

21. M. Arulampalam, S. Maskell, N. Gordon and T. Clapp “A tutorial on particle filters for online nonlinear/non-gaussian Bayesian tracking.” IEEE Trans. Signal Proc., 50, 2, 174-188, 2002.

22. P. Djuric, J. Kotecha, J. Zhang, Y. Huang, T. Ghirmai, M. Bugallo and J. Miguez, “Particle Filtering.” IEEE Signal Proc. Mag. 20, 5, 19-38, 2003.

23. S. Haykin and N. de Freitas, “Special Issue: Sequential state estimation: from Kalman filters to particle filters.” Proc. IEEE, 92, 3, 399-574, 2004.

24. A. Doucet and X. Wang, “Monte Carlo methods for signal processing,” IEEE Signal Proc. Mag. 24, 5, 152-170, 2005.

25. J. Candy, “Bootstrap particle filtering for passive synthetic aperture in an uncertain ocean environment.” IEEE Signal Proc. Mag. 24, 4, 63-75, 2007.

26. S. Kay, Modern Spectral Estimation: Theory and Applications (Englewood Cliffs, NJ: Prentice-Hall, 1988).

27. S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Englewood Cliffs, NJ: Prentice-Hall, 1993).

28. S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Englewood Cliffs, NJ: Prentice-Hall, 1998).

29. C. Therrian, Random Signal Processing: Detection Theory (Englewood Cliffs, NJ: Prentice-Hall, 1991).

30. R. Brown and P. Hwang, Introduction to Random Signals and Applied Kalman Filtering (New York: Wiley, 1997).

31. A. Jazwinski, Stochastic Processes and Filtering Theory (New York: Academic Press, 1970).

32. A. Sage and J. Melsa, Estimation Theory with Applications to Communcations and Control (New York: McGraw-Hill, 1971).

33. A. Gelb, Applied Optimal Estimation (Cambridge, MA: M.I.T. Press, 1974).

34. B. Anderson and J. Moore, Optimum Filtering (Englewood Cliffs, NJ: Prentice-Hall, 1979).

35. P. Maybeck, Stochastic Models, Estimation and Control (New York: Academic Press, 1979).

36. M. S. Bozic, Linear Estimation Theory (Englewood Cliffs, NJ: Prentice-Hall, 1979).

37. T. Kailath, Lectures on Kalman and Wiener Filtering Theory (New York: Springer-Verlag, 1981).

38. T. Kailath, A. Sayed and B. Hassibi, Linear Estimation (Englewood Cliffs, NJ: Prentice-Hall, 2000).

39. J. Mendel, Lessons in Estimation Theory for Signal Processing, Communications, and Control (Englewood Cliffs, NJ: Prentice-Hall, 1995).

40. M. Grewal and A. Andrews, Kalman Filtering: Theory and Practice (Englewood Cliffs, NJ: Prentice-Hall, 1993).

41. Y. Bar-Shalom and X. Li, Estimation and Tracking: Principles, Techniques and Software (Boston: Artech House, 1993).

42. D. Simon, OptimalState Estimation: Kalman, Hand Nonlinear Approaches (Hoboken, NJ: Wiley, 2006).

43. B. Widrow and S. Stearns, Adaptive Signal Processing (Englewood Cliffs, NJ: Prentice-Hall, 1985).

44. S. Orfanidis, Optimal Signal Processing (New York: MacMillan, 1988).

45. L. Sharf, Statistical Signal Processing: Detection, Estimation and Time Series Analysis, (Reading, MA: Addison-Wesley, 1991).

46. S. Haykin, Adaptive Filter Theory (Englewood Cliffs, NJ: Prentice-Hall, 1993).

47. M. Hayes, Statistical Digital Signal Processing and Modeling (Hoboken, NJ: Wiley, 1996).

48. P. Stoica and R. Moses, Introduction to Spectral Analysis (Englewood Cliffs, NJ: Prentice-Hall, 1997).

49. L. Ljung, System Identification: Theory for the User (Englewood Cliffs, NJ: Prentice-Hall, 1987).

50. L. Ljung and T.Soderstrom, Theory and Practice of Recursive Identification (Cambridge, MA: M.I.T. Press, 1983).

51. G. Goodwin and K. S. Sin, Adaptive Filtering, Prediction and Control (Englewood Cliffs, NJ: Prentice-Hall, 1984).

52. J. Norton, An Introduction to Identification (New York: Academic Press, 1986).

53. T. Soderstrom and P. Stoica, System Identification (New York: Academic Press, 1989).

ACKNOWLEDGMENTS

My wife, Patricia, was instrumental in suggesting that I write this text, so a big hearty thank you, with love to Patricia. My mother, Anne, who taught me the important lesson of tenacity, is an influence that will continue forever, thanks Mom. Of course, the constant support of my great colleagues and friends, especially Drs. E. Bond, G. Clark and M. Krnjajic, who carefully reviewed the manuscript and suggested many improvements, as well as Drs. D. Chambers, E. Sullivan, S. Godsill, N. Kingsbury and W. Ng cannot go without a strong acknowledgement. I wish to acknowledge the support and motivation from Dr. Simon Haykin, a colleague and friend.

1

INTRODUCTION

1.1 INTRODUCTION

In this chapter we motivate the philosophy of Bayesian processing from a probabilistic perspective. We show the coupling between model-based signal processing (MBSP) incorporating the a priori knowledge of the underlying processes and the Bayesian framework for specifying the distribution required to develop the processors. The idea of the sampling approach evolving from Monte Carlo (MC) and Markov chain Monte Carlo (MCMC) methods is introduced as a powerful methodology for simulating the behavior of complex dynamic processes and extracting the embedded information required. The main idea is to present the proper perspective for the subsequent chapters and construct a solid foundation for solving signal processing problems.

1.2 BAYESIAN SIGNAL PROCESSING

The development of Bayesian signal processing has evolved in a manner proportional to the evolution of high performance/high throughput computers. This evolution has led from theoretically appealing methods to pragmatic implementations capable of providing reasonable solutions for nonlinear and highly multi-modal (multiple distribution peaks) problems. In order to fully comprehend the Bayesian perspective, especially for signal processing applications, we must be able to separate our thinking and in a sense think more abstractly about probability distributions without worrying about how these representations can be “applied” to realistic processing problems. Our motivation is to first present the Bayesian approach from a statistical viewpoint and then couple it to useful signal processing implementations following the well-known model-based approach [1, 2]. Here we show that when we constrain the Bayesian distributions in estimation to Markovian representations using primarily state–space models, we can construct sequential processors capable of “pseudo real-time” operations that are easily be utilized in many physical applications. Bayes’ rule provides the foundation of all Bayesian estimation techniques. We show how it can be used to both theoretically develop processing techniques based on a specific distribution (e.g., Poisson, Gaussian, etc.) and then investigate properties of such processors relative to some of the most well-known approaches discussed throughout texts in the field.

Bayesian signal processing is concerned with the estimation of the underlying probability distribution of a random signal in order to perform statistical inferences [3]. These inferences enable the extraction of the signal from noisy uncertain measurement data. For instance, consider the problem of extracting the random variate, say X, from the noisy data, Y. The Bayesian approach is to first estimate the underlying conditional probability distribution, Pr(X|Y), and then perform the associated inferences to extract , that is,

where the caret, denotes an estimate of X. This concept of estimating the underlying distribution and using it to extract a signal estimate provides the foundation of Bayesian signal processing developed in this text.

Let us investigate this idea in more detail. We start with the previous problem of trying to estimate the random parameter, X, from noisy data Y =y. Then the associated conditional distribution Pr(X|Y =y) is called the posterior distribution because the estimate is conditioned “after (post) the measurements” have been acquired. Estimators based on this a posteriori distribution are usually called Bayesian because they are constructed from Bayes’ rule, since Pr(X|Y) is difficult to obtain directly. That is,

(1.1)

where Pr(X) is called the prior distribution (before measurement), Pr(Y|X) is called the (more likely to be true) and Pr(Y) is called the (scales the posterior to assure its integral is unity). Bayesian methods view the sought after parameter as random possessing a “known” distribution. As measurements are made, the is transformed to the function adjusting the parameter estimates. Thus, the result of increasing the number of measurements is to improve the distribution resulting in a sharper peak closer to the true parameter as shown in

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