PREFACE TO FIRST EDITION

In the real world, systems designed to extract signals from noisy measurements are plagued by errors evolving from constraints of the sensors employed, 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 nonGaussian 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], 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's perspective 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 equivalently 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...