Uncertainty is fundamental and unavoidable in the world. In this impressive mathematical book, the author uses a distinct approach to represent and examine uncertainty. He presents a meaningful and unified formal framework for uncertainty, using logic, probability theory, and common-sense linguistic terms. In doing so, he takes the study of uncertain reasoning in directions that are comprehensive, rich, and remarkable. The book contains 12 chapters, references, a glossary, and an index. It will be a good resource for students, logicians, philosophers, statisticians, and economists. It could also be a very useful textbook, offering a unique insight on uncertainty.
Chapter 1 introduces examples of uncertainty, and presents a brief overview of the book. Chapters 2 and 3 introduce representation issues for uncertainty. Chapter 2 introduces sets of probability measures for uncertainty, including Dempster-Shafer belief functions, possibility measures, and ranking functions. The author argues that these numeric representations are not enough to represent the nonnumeric likelihood of events, and then introduces plausibility measures as a generalized approach. The idea is to map a measured algebra set into an arbitrary partially ordered set. The advantage of this generality is that it is then possible to prove general results about representations of uncertainty that have property X; they also have property Y. The intensive exercises and notes in this chapter help to provide a deep understanding of representations of uncertainty. Chapter 3 is an extension of the concepts in chapter 2 in which new information depends on other new information that should be taken into account. Each method of representing uncertainty considered in chapter 2 is updated with this knowledge of conditioning.
Chapter 4 describes an independence notion that has been studied in depth in the context of probability. Bayesian networks are introduced to describe likelihood measures. Qualitative Bayesian networks, also called belief networks and defined by a directed acyclic graph, are able to reason about the relationships between events. This is an ideal graphic representation for probabilistic information measures.
Chapter 5 explains the expectation that measures the average value of the variable. The author discusses the expectations of those concepts, developed in chapters 2 and 3. To help agents make a rational decision, decision theory is introduced. The decision problem requires certain characteristics and belief measures, using a utility function and plausibility measure.
Chapters 6 and 7 cover reasoning frameworks and logics. Chapter 6 introduces three frames. Epistemic frames involve reasoning with conditioning knowledge. The advantages of epistemic frames are presented as a labeled graph. Probability frames are another kind of reasoning frame. These are a naturally placed constraint on probability assignments. Both frames are static. Multiagent systems are then introduced to provide structured reasoning frames that incorporate a time factor. These frames can perform distributed computation. Protocols are developed for representing the certainty of agent behavior. Chapter 7 presents formal logics for reasoning about uncertainty. The text includes perfectly applied semantic models for the various types of frames discussed in chapter 6. Proposition logic, modal logic, and theorems and lemmas are developed for reasoning about probability, likelihood, knowledge, and expectation.
Chapter 8 paves the way for reasoning about beliefs and counterfactuals. Default reasoning reaches toward conclusions, while counterfactuals reach a conclusion that is counter to fact, with assumptions. This chapter explains how probability measures and plausibility measures are used to give semantics to default and counterfactual reasoning.
Conditioning, introduced in chapter 3, is now extended to probability measures. Chapter 9 is on belief revision, and discusses conditioning probability measures as a representation of belief. The circuit-diagnosis problem is used as an example to explain the issues involved in belief revision. This is the most advanced topic in this book.
Chapter 10 introduces first-order modal logic. First-order logic can express relations and functional connections between individuals. The chapter also explains how first-order logic can be used for reasoning knowledge, probability, and condition logic. I think this chapter would be better placed after chapter 8, so that the reasoning logics can be introduced together.
Chapter 11 is another explanation of beliefs, from a statistic point of view. This concept is very important in practice. The author points out that there is no definitive right way to relate statistical information to degrees of belief. Therefore, the random-worlds approach is an easy way to explain and understand possible worlds, and to interpret default reasoning.
The summary, in chapter 12, is an overview of key points in the book. The author proposes a general representation of uncertainty, and develops a reasoning framework and logic. He raises a question about whether or not finding an easy representation of uncertainty is still an issue. He believes that this issue will be solved, both in theory and in practice.
A list of over 350 references, through 2002, ends the book. It includes not only a comprehensive list of sources on uncertainty, but also pointers to further information on reasoning about uncertainty.
The book contains deep mathematic insights, which are presented in an accessible context. For students, this is a must-read introduction to uncertainty. Other professionals will find the author’s unique insight a rich source of inspiration in solving uncertain problems.
