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Reviewer: Randy L. Garrett

While the focus of this graduate-level textbook is on causal networks, it discusses a number of approaches to adding reasoning under uncertainty to expert systems. It deals with numerically representing uncertainty in expert systems when the uncertainty as among members of a set of mutually exclusive and exhaustive alternatives. Propositions are considered to be either completely true or completely false; numbers are used to represent the expert system's uncertainty about the proposition's verity. Humans do not ordinarily reason numerically. They do, however, reason in terms of dependencies and conditional independencies as learned from the event frequencies that they experience. The author shows how to encapsulate some of this human ability into mathematical formalisms. The first chapter is a brief review of expert systems, mathematical logic, set theory, and uncertain reasoning. Considering that all this material is covered in less than 25 pages, it can only serve to refresh one's memory. An introduction that starts at the foundations and quickly builds to the real topics is typical of textbooks of all kinds, but especially mathematical texts. This wastes time and is necessarily short since an entire book might be written on each section, but it does provide a foundation for further discussion by defining terms and notations that often differ from author to author. The second chapter contrasts several approaches to probability. A discussion of different interpretations of probability theory is reasonable because different types of knowledge might require different interpretations of what probability is. Probability has both physical and philosophical foundations. This text considers three means by which probabilities may be assigned. The classical and most common method uses Laplace's principle of indifference: “alternatives are always judged to be equiprobable if we have no reason to expect or prefer one over the other.” A second method is statistical; probabilities are based on their relative frequencies of occurrence. This method is difficult to use if no obvious frequency basis exists. The third method is to assign probabilities solely on individual beliefs. Since this list is not mutually exclusive or exhaustive, the theories presented in this book do not allow us to choose between these methods. The review continues into the third chapter with a discussion of graph theory, which will provide a theoretical foundation for the consideration of causal networks. The fourth chapter compares causal networks to rule-based systems of representing uncertainty. Rule-based systems face a number of difficulties in propagating probability values. When the probability of underlying assertions changes, calculating the change in the probability of derived conclusions can be difficult. As a consequence, a number of ad hoc methods have been constructed. Causal networks, also called Bayesian or belief networks, can simplify probability propagation under certain circumstances. The author constructs a mathematical foundation for causal networks based on directed acyclic graphs (DAGs). He then considers the implications of causal networks in encapsulating human understanding of probabilities and discusses mechanisms for incorporating causal networks into expert systems. A causal network is created by constructing a DAG in which the directed arc between vertices corresponds to human understanding of cause and effect. Arcs go only from direct causes to effects. The notion of causality is not technically required, however—conditional independencies are sufficient—so the model can be extended if required. Next, Neapolitan considers probability propagation in increasingly complicated networks. Beginning with singly connected networks, he proceeds to trees of cliques. Singly connected networks are handled with a local probability propagation scheme developed by Pearl. The author discusses extensions to trees and non-singly connected networks. The latter may be dealt with using any of three techniques: conditioning, stochastic simulation, or clustering. In addition to developing considerable mathematical underpinnings, the author provides algorithmic procedures and a few examples that illustrate how to calculate probabilities. The algorithms are rather complex, unfortunately, and not computationally efficient. In fact, probability propagation for many networks has been proven to be NP-hard. This means that an efficient general- purpose algorithm for probability propagation in large arbitrary networks is unlikely to be developed, leaving us with stochastic simulations, approximation algorithms, or heuristics. Next, Neapolitan discusses abductive inferences (reasoning processes that derive the best explanation for a given set of evidence) in both singly and arbitrarily connected networks, which are a common use of expert systems for diagnostic problems. He derives extensive mathematical proofs along with appropriate algorithms for implementation. All the algorithmic procedures in this book are high level, however, and will require some work to translate into working computer programs. The author also considers how to use the probabilities derived using causal networks to arrive at recommended decisions. He does this by using decision trees and causal networks. The book concludes by considering how to obtain the values to assign to probabilities. In the absence of statistical studies, this can be the most difficult part of actually adding reasoning under uncertainty to an expert system. Humans are generally not good at assigning conditional probabilities, and many experts are reluctant to do so. The text is highly mathematical in tone with a somewhat uneven flow of discourse and concepts. The author freely expresses his opinions. The book does, however, span most of the field of probabilistic reasoning and contains quite a bit of useful information and a number of interesting examples. A thorough set of references is provided, but a more complete development of algorithmic examples would have been helpful for the practicing expert system builder.