This book consists of 16 closely interrelated chapters. It starts by introducing the basic concepts of the theory (Chapters 1 to 6), then turns to general fields of application (Chapters 7 to 11), provides an insight into the use of imprecise probabilities in engineering, financial risk measurement and reliability analysis (Chapters 12 to 14) and concludes with aspects of elicitation and computation (Chapters 15 and 16).
The notion of desirability, as described in Chapter 1, lies at the core of the behavioural approach to imprecise probabilities, and the discussion there provides a foundation and justification for the account of lower previsions presented in later chapters. It deals with coherence, and with the associated notion of inference using natural extension, marginalization and conditioning.
Chapter 2 discusses (conditional) lower previsions and the notions of coherence associated with them. It also shows how the notion of natural extension provides the theory with a powerful method of conservative (or least committal) probabilistic inference.
Chapter 3 explains how to deal with structural assessments of independence and symmetry in the theory of coherent lower previsions. In particular, the different concepts of independence used in later chapters are explained in detail.
Chapter 4 investigates simplified representations by less general model classes, which are often useful as they are easier for computation and elicitation, and because the full generality of the theory of lower previsions is not always required.
Chapter 5 discusses, within the context of imprecise probabilities, a variety of ways of looking at uncertainty modelling in general, including common constructive elements as well as discrepancies and differences.
Chapter 6 offers a look at lower and upper previsions (or prices) with an interpretation that is rather different from the behavioural one introduced in Chapter 2. The authors discuss their game-theoretic approach to probability, which has its origins in the seminal work of Jean Ville on martingales.
Chapter 7 gives an introduction to the powerful use of imprecise probabilities in statistical inference. In particular it discusses generalized Bayesian inference, extensions of frequentist testing and estimation theory, including their relation to robust statistical methods, as well as nonparametric predictive inference, and provides some insight into modelling of data imprecision.
Chapter 8 discusses nonsequential and sequential decision making under imprecise probability. In particular, the main criteria generalizing expected utility are introduced.
Chapter 9 provides an overview of the powerful use of imprecise probabilities in probabilistic graphical modelling, where complex models over a possible large set of variables are built by combining several submodels. The construction of such credal networks and their updating is discussed.
Chapter 10 describes imprecise probability methods for classification, i.e. predicting the class of an object on the basis of its attributes. This is an application area where the paradigmatic shift to imprecise probability has proven to be particularly powerful, and its success has also been corroborated by large scale comparison studies. Imprecise probabilities allow to overcome major difficulties of traditional methods, notably prior dependency of traditional Bayesian classifiers and overfitting and instability of classification trees.
Chapter 11 gives a first introduction to stochastic processes under imprecise probabilities. The chapter discusses the event-driven approach to stochastic processes and its generalization. It then focuses on the practically very important case of imprecise Markov chains. It compares models based on different independence concepts and investigates the limit behaviour of imprecise Markov chains.
Chapter 12 presents strong arguments for the need to use imprecise previsions in finance. It is shown that theory of imprecise probabilities provides many tools that are closely linked to popular concepts in financial risk measurement, and it enables modelling based on fewer and simpler assumptions than the standard approaches.
Chapter 13 illustrates several engineering applications in which the opportunity to use weaker modelling assumptions underlying structural models fits neatly with the lack of perfect information about certain aspects in reality. Links to sensitivity analysis and practical decision making in engineering are also discussed.
Chapter 14 provides examples where the theory of imprecise probabilities provides natural problem formulations in reliability and risk analysis. Problems of stress-strength reliability are formulated as constrained optimization problems, with the constraints reflecting the limited information available. Some statistical methods for reliability and risk problems are also discussed, including inference about unobserved or even unknown failure modes.
Chapter 15 gives an introduction to the important but relatively unexplored topic of elicitation of imprecise probability models, looking at methods for evaluating judged probabilities and influence factors on probability judgements.
Chapter 16 briefly discusses a number of computational aspects of imprecise probabilities, focusing on natural extension and decision making in particular.