There exists no scientific knowledge without some degree of uncertainty concerning its validity, reliability, or precision. Probability is our most important construct to deal with this uncertainty. Although uncertainty exists in all fields of scientific activity, the standard of dealing with it varies from subject to subject, depending mostly on the particular way in which scientific knowledge is obtained and formulated, but also on traditions. This Handbook sets the ambitious goal of presenting the fundamentals and several applications of probabilistic thinking in the social and behavioral sciences, economics, and law. Elements of probability are included in most relevant university curricula, but very often probability is discussed as background material for statistics only, generating the impression that statistics itself is nothing else than the application of existing statistical methods. Just like the role of careful statistical modeling, as opposed to routine application of statistical tools, is often suppressed, the presentation of probabilistic thinking is also often reduced to the static presentation of simple rules of the calculus of probability.

In fact, probability is a lively subject that develops very fast, and some understanding of it is necessary for the correct interpretation of any scientific finding. The chapters of the Handbook cover different aspects of probability. Some are philosophical, some are mathematical, and some are statistical in their nature. The diversity of the approaches to probabilistic thinking is well represented by the diverging views of what is probability and how it is related to reality. The authors of Chapters 2 and 3 agree that probability is a construction that helps us conceptualize uncertainty, but Chapter 2 takes the view that observable events occur with a certain probability that may be revealed by repeated observations (the so-called frequentist definition of probability), while Chapter 3 discusses probability related to subjective judgment and expectation (the so-called subjective probability). On the other hand, Chapter 4 illustrates that not all subjective interpretations of probability are consistent. Closely related to probability is the concept of randomness: An event is random if the observer has uncertainty with respect to it. The observer may choose a deterministic or a stochastic model to describe the behavior of the possible observations, and the choice is often dictated by the available knowledge and other resources, but very often it also has an element of free choice (Chapter 2). On the other hand, Chapter 9 attributes “real” randomness to certain physical processes and takes the perhaps surprising but very useful position that processes that cannot be distinguished from a real random process with a reasonable amount of effort may be considered random. Furthermore, Chapter 6 distinguishes between aleatory and epistemic uncertainty (to refer to a kind of impossibility to remove uncertainty and to the actual lack of knowledge to do so). These issues are also related to the debate between the frequentist and the Bayesian views on statistical analysis (discussed in Chapters 5 and 6). Yet another possibility of handling uncertainty by using fuzzy sets is mentioned (in a particular context) in Chapter 20. A related topic that has attracted significant scientific interest recently and is taken up in more than one chapter is the definition and analysis of causality in stochastic models—that is, when causal relationships have to be identified in the presence of uncertainty. Chapters 8 and 23 discuss the most important approaches, one through the definition of individual causal effects and the other by the application of graphical models. There are several other topics discussed in more than one chapter, including time-dependent observations, measurement of and protection from investment risk, and DNA evidence.

While some of the chapters cover standard textbook material in a concise way and some others tend to describe cutting-edge developments, the presentation is intuitive throughout and the mathematical details are suppressed, so that all chapters are accessible with good high school or first-year college mathematics. The Handbook largely neglects the calculus of probability: Such details are left to specialized textbooks, and it concentrates mostly on the conceptual development of probabilistic thinking and on many of its applications. Most of the chapters are self-contained, which implies that some material is discussed in more than one chapter, perhaps from a different perspective, as was illustrated above. The references given at the end of each chapter vary from mentioning the few most fundamental sources to listing a complete collection of research papers, depending on the material covered in the chapter.

The Handbook is divided into three parts. The first part covers the theory of probability and gives an introduction to its mathematical and philosophical aspects in a fairly nontechnical way but well beyond the level at which most social or behavioral scientists study this topic at university. Chapter 1, written by Peter Lee, presents the most important developments in the history of probability theory. Many of these results are discussed and explained in detail in later chapters. Chapter 2, written by Herwig Friedl and Siegfried Hörmann, summarizes the main results of probability theory, as a mathematical subject. They give an intuitive but precise description of some of the more involved theorems, including the law of large numbers and the central limit theorem. Chapter 3, written by Igor Kopylov, gives an overview of the subjective definition and interpretation of probability. The chapter shows that many of the fundamental properties of probability, which were derived in the previous chapter in a frequentist setting (as this leads most easily to these results), may also be obtained from a different approach to the relationship between reality and probabilistic thinking. Chapter 4, written by Nicholas Shackel, takes a critical view on assumptions and beliefs often associated with probability and shows many of these to be untenable. These paradoxes occur, of course, not within the mathematical theory of probability but rather in our thinking when we try to formulate probabilistic models to describe reality.

The second part extends the theory given in the first part and discusses the general ways in which probabilistic approaches are used in research. The coverage here concentrates on applied probability and theoretical statistics. Chapter 5 is an account of the standard (frequentist) way in which probability theory is applied in statistics. Probability theory tells us how random observations from a population (“the reality”) tend to behave, depending on the characteristics of this population and the sampling procedure, and statistics uses this knowledge to infer from observations to the characteristics of the population whence they were selected. Chapter 6, written by Tony O'Hagan, gives an account of Bayesian statistics and compares it with the frequentist approach. The main difference is that in frequentist statistics the characteristics of the population from which the observations came are assumed to be fixed but unknown, while in Bayesian statistics uncertainty regarding population characteristics is allowed (and is described probabilistically). However modest this difference appears to be, it leads to wide-ranging deviations in terms of the questions that may be asked and also in the ways in which those questions are answered. Chapter 7, written by Mauro Gasparini and Maria Piera Rogantin, describes experiments—one of the most important ways in which researchers collect information with respect to reality. The design of experiments is fundamental in technical applications of statistical analysis, but the chapter, in addition to defining the general concepts of main effect and interaction, emphasizes conceptual aspects and applicability to human populations. Chapter 8, written by Michael Sobel, gives an account of one of the most exciting areas of probabilistic thinking these days: how causal effects may be defined, observed, and tested in the presence of uncertainty. The chapter summarizes the ongoing scientific debate on this topic and also gives practical illustrations of the most important analyses. Chapter 9, written by Oded Goldreich, presents yet another approach to defining and handling randomness through computational complexity. This approach proves very useful not only in apparently theoretical efforts, such as algorithmic theorem verification, but also in related practical applications, such as cryptography.

The third part considers several applications of probabilistic modeling. Much of the content of the third part describes methods and uses concepts from statistics, but the focus here is not on estimation or testing, although these aspects are also discussed, but rather on model building with the aim of incorporating uncertainty. Chapter 10, written by Michael Lewis, describes the fundamentals of time-series analysis. This approach is used when one has reason to believe that phenomena observed over time are governed not only by the actual status of the population where the observations came from but also by earlier statuses of the same population. Chapter 11, written by Nancy Brandon Tuma, discusses another aspect of the temporal nature of observations, when the information recorded describes the occurrence of certain events of interest. This approach, survival analysis, also referred to as event history analysis, deals with modeling the factors that influence the probability of a certain event occurring within a specific period of time. Chapter 12, written by Jeffrey Wooldridge, deals with sampling and the implications of the sampling procedure for estimation and testing. The discussion goes way beyond elementary simple random sampling and, among other topics, proposes methods for modeling nonresponse. Chapter 13, written by Edward Frees and Jee-Seon Kim, presents another approach to modeling observations over an extended period of time. Particular attention is paid to the advantages and disadvantages of data arising from repeated observations of the same sample (the panel) and to temporal aspects that may be relevant to explain cross-sectional associations. Chapter 14, written by Vasja Vehovar, Metka Zalatel, and Rudi Seljak, deals with probabilistic methods in official statistics. Official statistics has played and continues to play an important role in the development of probabilistic methods for surveys, and the chapter describes the designs of the most frequently carried out surveys by national statistical offices and discusses related issues such as small-area estimation, data fusion, and seam effects. Chapter 15, written by Nick Longford, discusses probabilistic models for measurement error and misclassification. The proposed method of inference considers the true value as a latent variable, and a general solution based on imputation is suggested. Chapter 16, written by Klaas Sijtsma and Wilco Emons, considers a very important case when inference is to be based on imprecise observations: probabilistic developments of tests, in particular for educational purposes, using item response theory. The chapter describes several models and illustrates their applications to real data. Chapter 17, written by Klaus Troitzsch, shows how probabilistic simulation methods may be used to model societal phenomena such as demographic processes or opinion formation. The discussion covers approaches of different complexity, including microsimulation and multi-agent models. Chapter 18, written by Philippa Pattison and Garry Robins, is about the rapidly developing topic of probabilistic network analysis. Probabilistic models are discussed for the evolution of social networks, and several ways for the probabilistic analysis of such networks are described, including simulation methods based on Markov Chain Monte Carlo. Chapter 19, written by Chas Friedman, gives insight into the topic of probabilistic analysis of gambling, a topic that played a very important role in the historic development of probability theory. Several games are investigated, and strategies that are optimal on the average are developed for these. Chapter 20, written by Richard Derrig and Krzyszt of Ostaszewski, describes how probabilistic approaches are applied in insurance. In addition to discussing the most important theoretical aspects of actuarial science, the chapter expands on several theoretical and practical problems of the insurance business. Chapter 21, written by Ad Feelders, discusses probabilistic methods for another financial problem, the scoring of credit applicants. Several models for determining the probability that someone will not pay back a loan are described, and special attention is given to the difficulties arising from the fact that observations are available only for those who were actually given a loan, leading to a special incomplete data problem. Chapter 22, written by Craig Rennie, shows how optimal investment portfolios are designed from risky securities, where risk is measured by the standard deviation of the return of the investment. Among other theories, the famous Black-Sholes option-pricing model is described. Chapter 23, written by George Luger and Chayan Chakrabarti, describes how probabilistic reasoning may be used in achieving automatic decisions. The chapter discusses expert systems and belief networks based on graphical models, showing how the Bayesian approach may lead to [Page xiv]simplified and efficient decisions. Chapter 24, written by Julia Mortera and Philip Dawid, discusses how uncertainty related to evidence should be correctly interpreted in criminal investigations. In addition to discussing common mistakes and misinterpretations, they also show how Bayesian networks may be used to interpret complex evidence, including DNA data. Chapter 25, written by Basil C. Bitas, describes practical aspects of the presentation of probabilistic arguments in the courtroom. Science should be interpreted so that it becomes relevant for the law, and in addition to the main aspect of this integration, the prevailing practice in several countries from all over the world is discussed.

An edited volume, like this one, is always the result of the effort of several contributors. First and foremost, I am indebted to all the authors for their contributions and also for their willingness to subject their manuscripts to the several rounds of the review procedure. Perhaps no editor could be equally well qualified in all the fields covered in the Handbook, and I learned a lot while working with the authors. I am also indebted to the members of the Advisory Board, who helped me through several stages of this project. Very special thanks go to Lisa Cuevas Shaw, former Acquisitions Editor at Sage, who first proposed the idea of such a Handbook and applied successful tactics to convince me that I should try to put it together. She also provided me with constant moral support throughout the two years of actual editorial work. I feel fortunate to have been able to work with Sage's production team and, in particular, with Melanie Birdsall and Shankaran Srinivasan, who not only masterly handled the technical aspects of book production but were also very accommodating when it came to consolidating the individual chapters into one volume.

Paul Barrett (Auckland, New Zealand) made a very special contribution to the Handbook by trying to write a chapter on probabilistic profiling in psychology (criminal profiling, executive recruitment, etc). After much effort, he came to the conclusion that very little probability was being actually used in these activities and suggested that we drop the chapter. I thank him for his effort, and perhaps, this could be a new and fruitful field for the application of probabilistic thinking.

My very sad duty is to let the readers of the Handbook know that Professor Chas Friedman (Austin, Texas), author of the chapter on gambling, passed away shortly after completing the manuscript of his contribution, at the age of sixty. Although I have never had a chance to meet Professor Friedman personally, during our collaboration I learned to appreciate his disciplined thinking and wide-ranging scientific interests. This chapter is likely to be his last publication.

The hope, shared, I am sure, by all contributors, that such a Handbook may not only serve as a source of reference but also influence the way scientific research is conducted by some of its readers helped us overcome many difficulties while working on the manuscript. When the work is close to coming to an end, the editor may only wish that many readers will find the effort worthwhile.