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Single photon emission computed tomography (SPECT) is a medical imaging technology. The patient is treated with a radioactive substance that releases individual gamma rays (in small amounts, so as to be detectable but not damaging to the patient). Some of these gamma rays are detected by a special camera, and then a classic inverse problem results from the need to infer the original image in the patient from the counts of detected gamma rays. Computer code links hypothesized images to predicted gamma ray counts. The primary goal is to reconstruct the features of the underlying object, such as a patient's brain. This image is typically desired at a particular resolution, and only one type of data is collected, so there is nothing inherently multiscale here. However, a fully Bayesian analysis requires the use of Markov chain Monte Carlo methods, and the chain can take a long time to run because of poor mixing. The use of multiscale techniques can greatly speed up the computational aspects of the problem, improving the statistical inference that can be done in a reasonable amount of time. Thus, we demonstrate the implicit methods of Part IV with this example. [ABSTRACT FROM AUTHOR]

Many important applications in the cleanup of contaminated soil and in the production of petroleum from oil fields require the understanding of soil structure and prediction of fluid flow patterns. Subsurface flow is dependent on a number of factors, with one of the primary ones being the permeability field. Permeability is a measure of how easily fluid moves through the porous medium at a point, and it varies spatially. In this chapter, we treat permeability as a constant at a point, but in more detailed applications it may be tensor-valued to account for the fact that soil structure can be directional and so fluid may flow more easily in some directions than in others. [ABSTRACT FROM AUTHOR]

This chapter starts by reviewing the key ideas of genetic algorithms and then demonstrates how to use them in a multiscale setting. The direct extension to multiscale modeling is useful for maximization. However, these multiscale ideas can also be extended to the Bayesian framework in combination with Markov chain Monte Carlo ideas. The resulting approach can then be seen as a generalization of the Metropolis-coupled methods of the previous chapter. This combination is a powerful method for fully Bayesian multiscale modeling, particularly in a parallel computing environment. Compared with the methods in the previous two chapters, the key extension of this chapter is to allow the swapping across scales (or "temperatures") of only part of the information in the chains instead of requiring the entire current state of the chain to move between scales. This adaptation can improve the probability of accepting a swap in complicated problems and thus further improve the mixing of the chains. [ABSTRACT FROM AUTHOR]

The methods of the previous chapter are useful for employing a multiscale approach for computational efficiency but are not as useful when one is faced with truly multiscale data. In contrast, the methods in this chapter are equally useful as a computational tool or for a full analysis of multiscale data. These methods can be seen as a generalization of simulated sintering, where instead of having a single chain moving between scales, multiple chains are run so that there is one (or more) chain at each scale at each point in time. These chains periodically exchange scales with each other, which allows information to move across scales but also ensures that we continue to sample at every scale. This eliminates some of the difficulties in posterior inference that can occur when trying to incorporate samples which are disjoint in time. In particular, simulated sintering produces a single MCMC run over the joint space of the parameters and the scale, but for any particular scale, there will only be MCMC samples at unconnected intervals. [ABSTRACT FROM AUTHOR]

In contrast to the rest of the multiscale models discussed in this book, the methods of this section do not attempt to explicitly model the joint distribution of the data at all of the scales. Instead, a probability model is specified for each scale independently, and then the scales are linked during the fitting algorithms. A fully dependent multiscale fit is then obtained implicitly. [ABSTRACT FROM AUTHOR]

In contrast to the grid-based methods of the previous chapters in this section, the general class of methods often referred to as "change of support" methods evolved mostly to deal with problems of data that are observed at a scale different from the scale at which inference is desired, and the mapping between scales is nonregular. For example, demographic data may be available by postal code, but information about particular city blocks might be of interest. Alternatively, one could be trying to relate such demographic data to voting data, which are reported by voting precinct, and these precincts may not have any relationship to postal codes (so that precincts may overlap multiple postal codes, and postal code regions may contain multiple voting precincts). How is one to use the available data to draw valid conclusions at a different scale? The desire to answer this question is the primary motivation for the change of support problem. [ABSTRACT FROM AUTHOR]

This chapter presents multiscale time series models that have been introduced by Ferreira et al. (2006). These models couple standard linear models at different time scales with linear link equations between scales. As in the multiscale random field models presented in Chapter 10, Jeffrey's rule of conditioning is used to ensure that the processes at the different levels are compatible. This allows consistent modeling of time series at different levels of resolution (e.g., daily or monthly aggregates of financial or meteorological data) and also allows coherent combination of information across time scales. [ABSTRACT FROM AUTHOR]

This chapter presents multiscale random field models with interconnected resolution levels and smoothness within each level that were introduced by Ferreira et al. (2005). This is important when interest lies in modeling the dynamics of the multiscale process within each scale of resolution as well as across the different scales. This is of particular interest when the process is observed at different scales of resolution or when the multiscale model is used as a prior for an underlying process, and it is desirable to impose smoothness within each level of resolution and a stochastic constraint that the different levels of resolution have similar behavior. [ABSTRACT FROM AUTHOR]

In many multiscale problems, the relationship between the different levels of the multiscale process is deterministic. For example, the total volume of petroleum in a given region at a coarser level is the sum of the volumes of petroleum in the corresponding finer regions. This type of property is known as balance of mass. In this chapter, we revise the tree modeling approach of Kolaczyk and Huang (2001) for the analysis of multiscale processes subject to mass balance. Their approach is strongly based on the approach of Allan S. Willsky and his coauthors, presented in Chapter 7, but generalized to cases where the number of descendants is not constant and to include the cases of Gaussian and Poisson observations. The case of Poisson observations is a generalization of the work of Kolaczyk (1999) for multiscale Poisson processes on dyadic trees. As in Willsky's tree model, the multiscale model of Kolaczyk and Huang induces rich covariance structures whose analytic expressions were derived by Louie and Kolaczyk (2004). [ABSTRACT FROM AUTHOR]

We present in this chapter hidden Markov models on trees. These models are generalizations of the more traditional one-dimensional hidden Markov models. Traditional hidden Markov models (HMMs) assume hidden states that can take discrete values and are connected through a one-dimensional Markov chain (for a good review on HMMs, see Scott, 2002). In these HMMs, the observations may be discrete or continuous and are conditionally independent given the hidden states. Analogously, hidden Markov models on trees (HMMTs) assume that the values of the latent label process at nodes of a given level are conditionally independent given the latent label process at the immediate coarser level. Moreover, HMMTs assume that the latent label process evolves on a tree in a construction analogous to that described in Chapter 7 for Gaussian processes on trees. [ABSTRACT FROM AUTHOR]

Many multiscale approaches have appeared in the engineering literature, with a focus on the development of coarser representations of the phenomenon of interest in order to obtain fast computational algorithms. With this objective in mind, a particularly effective multiscale framework was developed by Allan S. Willsky and his coauthors in the 1990s (Basseville et al., 1992a,b; Luettgen et al., 1993; Chou et al., 1994a,b; Luettgen et al., 1994; Luettgen and Willsky, 1995a,b; Irving et al., 1997; Frakt and Willsky, 1998; Daoudi et al., 1999). [ABSTRACT FROM AUTHOR]

Multiscale modeling arises in a wide variety of applications. As discussed in Chapter 1, there are at least three classes of problems that can be modeled most effectively within a multiscale framework. In the first type, data are observed at different spatial scales and the model is used to integrate the information from the different scales. In the second type, data are observed only at the finest scale and the model is used to induce a particular process at that scale. In the third type, the observed data are related nonlocally and nonlinearly to an underlying multiscale process, and the model is used as a prior for that process. [ABSTRACT FROM AUTHOR]

Wavelets are one of the most well-known multiscale techniques and have been extensively explored in statistical applications in the 1990s and the 2000s. Wavelets are powerful mathematical tools that have been used with success in the compression of images and in the analysis of signals. They decompose signals and images in components spatially localized at different levels of resolution known as wavelet coefficients. [ABSTRACT FROM AUTHOR]

Standard Gaussian process models were introduced in Section 2.2. Such models gained popularity originally in geostatistical applications but have become widely used throughout spatial statistics. In this chapter, we will see how one can represent a Gaussian process via convolutions, which can have great computational advantages as well as providing a rather natural mechanism for moving to a multiscale model. Thus the latter section of this chapter will fully explore how to use the convolution representation to build a multiscale model. [ABSTRACT FROM AUTHOR]

Before launching into the main topics of the book, we first want to introduce two standard models used for spatial data, as they will reappear throughout the book. The first is the Markov random field (MRF), which is most useful for grids and irregular areal data. The second is the Gaussian process, which is more useful when a continuous surface is desired or a wider variety of spatial smoothness needs to be specified or fit. [ABSTRACT FROM AUTHOR]

We start from the view that the purpose of statistical theory is to explain and guide what we choose to present as statistical evidence, i.e., what data and reasoning should cause us to adopt certain conclusions. We arrive at the position that there are kinds of statistical evidence, each of which lends a different explanatory insight and none of which is perfect. The explanatory models which we consider—along with their criteria—appear in Table 14.1. [ABSTRACT FROM AUTHOR]

On the inside of the front cover of each issue of its Journal, the American Statistical Association declares that it is a scientific organization. In what sense is statistics scientific? There are several common usages of the word "science." First, science is sometimes used as a synonym for systematized knowledge. Or, in more detail, a science is the systematized knowledge produced by the study of the structure of a class of concepts. A second usage is that science is explanation. Third and more narrowly, science is sometimes taken to mean the systemized knowledge of "nature," of the "real world." A fourth usage, due to Karl Pearson, is that any field of study which employs the scientific method of hypothesis, deduction and experiment is a science. A fifth usage is common in physics; Ruhla (1993) writes "prediction = science." [ABSTRACT FROM AUTHOR]

As discussed in Chapter 8, Birnbaum introduces Ev (E, y), the evidential meaning of obtaining data y as an instance of experiment E. Following Birnbaum, various authors have wrestled with the problem of developing a single set of postulates under which statistical inference can be made coherent. But as we claim in Section 8.3, Ev(E, y)does not exist. Evidence is grounds for belief—an imprecise concept. There must be many valid reasons for believing and hence many ways of making the evidence concept precise. Most of our beliefs are held because mother—or someone else we trust—told us so. The law trusts sworn testimony. Scientific and statistical evidence are other different grounds for belief—supposedly particularly reliable kinds. Instead of Ev (E, y) we are concerned with Ev(E, T, y), the evidential meaning of observing v as an instance of E, in the context of theory T. [ABSTRACT FROM AUTHOR]

The use of significance tests to express statistical evidence has a long history. [ABSTRACT FROM AUTHOR]

As a consequence of Wald's (1950) powerful work, statistics was for a time defined as the art and science of making decisions in the face of uncertainty. The decision problem assumes the questionable true value model of Section 8.1 and contemplates deciding, on the basis of data, between various possible actions when the state of nature is unknown. It is anticipated that the data will be helpful in choosing an action since the probabilities of data depend on the state. The task of the decision maker is to choose a decision rule specifying the action d(y) to be taken if data y is observed. The theory assumes known numerical losses l (a, θ) in taking each action a when each state θ "obtains." [ABSTRACT FROM AUTHOR]

Rubin (1984) describes statistical inference to be Bayesian, if known as well as unknown quantities are treated as random variables—knowns having been observed but unknowns unobserved—and conclusions are drawn about unknowns by calculating their conditional distribution given knowns from a specified joint distribution. [ABSTRACT FROM AUTHOR]

We begin our discussion of statistical evidence by clarifying terms; while it won't quite serve as a definition, that aspect of statistics which we consider is concerned with models of inductive rationality. We will be concerned with rational reasoning processes which transform statements about the outcomes of particular experiments into general statements about how similar experiments may be expected to turn out. The purpose of statistical theory is to guide and explain statistical methods. [ABSTRACT FROM AUTHOR]

The idea of a probabilistic experiment or chance situation was introduced by example in Section 5.2. There are several attitudes concerning the indeterminism which we call chance: (i) chance is the opposite of determinism; (ii) the situation is in principle completely determinable but the initial conditions have been incompletely specified; (iii) in some respects the world is fundamentally random; and (iv) the indeterminism is located in the mind(s) of the observer(s) rather than in the exterior world. We shall immediately discuss (i) and (ii). The third attitude may be correct, but it begs for an explanation which we cannot provide. Discussion of attitude (iv) was begun in Chapter 6 and will continue. [ABSTRACT FROM AUTHOR]

Consider again the origins of our subject. The formulation of the de Mere-Pascal-Fermat problem of points was not in terms of equally likely or frequency or personal degree of belief but as a question concerning the value of a player's position in the game. The issue in general is, if one is to receive a prize or payment, but the amount of that payment is uncertain, then how does that uncertainty affect the value of the prize? A modern development is that this question has been answered in terms of a personal price interpretation of probability. This chapter is our version of the details. [ABSTRACT FROM AUTHOR]

Gambling is the origin of the mathematical theory of probability. Cardano, who died in 1576, wrote a 15-page "gambler's manual" which treated dice and other problems; however, the effective beginning of the subject was in the year 1654. The Chevalier de Mere was concerned over the following problem of "points": A game between two persons is won by the player who first scores three points. Each of the participants places at stake 32 pistoles, the winner taking the entire stake of 64 pistoles. If the two leave off playing when the game is only partly finished, then how should the stakes be divided? For example, if the players have one and two points, respectively, and their chances for winning each point are equal, then what should be the division of stakes? The Chevalier consulted Blaise Pascal, who solved the problem and communicated his solution to Fermat. In the ensuing correspondence, the two mathematicians initiated the study of probability theory. In Chapter 6 we shall return to the modern theory of gambling. [ABSTRACT FROM AUTHOR]

Like mathematics, statistics, too, is a handmaiden of science. See the preface in Senn (2003) in this regard. But, perhaps unlike mathematics, statistics derives much of its character and meaning from related disciplines. In addition to logic, mathematics, and scientific method, there are other bodies of thought to which statistics can look for guidance and with which it need come to terms. Further important guides are law, learning theory, and economics. Here we mention the first two; later, in discussing the price interpretation of probability, we touch on economics. [ABSTRACT FROM AUTHOR]

Evidence for the special success of science can be seen everywhere: in the automobiles we drive, the television we view, the increase in life expectancy, and the fact that man has walked on the moon. We may question the values, or lack thereof, of science, but surely science has been singularly successful in many of the enterprises which it has attempted. It is therefore a little surprising that what constitutes meaningful scientific work is one of the most hotly debated issues of our time. The nature of scientific process and method, the aims or goals of science, the nature and objectivity of scientific knowledge—all are controversial. Still, there is much agreement. This chapter discusses the controversy and attempts to find points of agreement. [ABSTRACT FROM AUTHOR]

It is a historical fact that geometry was crucial in the development of the modern view of mathematics and the axiomatic method. David Hilbert (1862-1943) judged that the invention of non-Euclidean geometry was "the most suggestive and notable achievement of the last century," a very strong statement, considering the advance of science during the period. Hilbert meant that the concepts of truth and knowledge, and how to discover truth and acquire knowledge, all this was changed by the invention of non-Euclidean geometry. This chapter reviews some of that historical development. [ABSTRACT FROM AUTHOR]

We begin by clarifying terms. Evidence is grounds for belief—an imprecise concept. There must be many valid reasons for believing and hence many ways of making the concept of evidence precise. Most of our beliefs are held because mother—or someone else we trust—told us so. The law trusts sworn testimony. Scientific and statistical evidence are other, different grounds for conclusion, supposedly particularly reliable kinds. [ABSTRACT FROM AUTHOR]