Review of Decisions under Uncertainty by Ian Jordaan: Cambridge University Press, Cambridge, U.K., 2005. ISBN: 13 978-0-521-78277-5. Price: $110.00. 672 pp.

This book takes a very interesting approach to the treatment of uncertainty and its role in decision making. The author, who has extensive background in both academia and industry, uses an intuitive argument whenever possible to derive common laws of probability. As such, he intends the book as “an introduction to probability for engineers” (quoted from the Preface), and appropriate for use in an engineering curriculum; exercises are included at the end of each chapter. This approach has a distinct advantage for such use in contrast to the more traditional mathematical foundation, from which most engineering students are discouraged by a perceived lack of applicability. Based on the author’s experience, it seems that the book would fit best into an undergraduate civil engineering curriculum, since applications are focused on decisions for the built environment. How the book fares in comparison to the two common probability texts oriented toward civil engineers, Probability, Statistics and Decision, by Jack Benjamin and Allin Cornell and Probability Concepts in Engineering Planning and Design, by Alfredo Ang and Wilson Tang, is unclear. Since these latter two were written about 3 decades ago, Jordaan’s book starts off with the advantage of current examples. Several of the topics in the book are normally considered only at the graduate level of engineering education (spectral analysis of random processes, optimization theory, and extreme value theory, etc.), and the inclusion of these may make the book somewhat daunting to undergraduate students. The stated goal that the book “will be of use to practicing engineers” is probably one more of hope than reality since the mathematics and details of examples quickly moves beyond the interest level of most professionals looking for answers. (For instance, the book makes the argument that “… probability is not a frequency… We use frequencies to estimate probabilities; this distinction is important.”) This may be true, but the importance and subtlety of the distinction is not expected to capture the attention and interest of practicing engineers.

This book encompasses a much broader array of topics than is typically covered in a single-semester course. In addition to traditional probability theory, it has extensive coverage of utility theory, which the author states is as fundamental as probability theory in the process of decision making (he refers to them as being “related dually”). The author also denotes significant space throughout the book to formal consideration of Bayesian methods, based on his evaluation that engineers are continually faced with the prospect of new information, which should be incorporated into their decision process. It remains to be seen whether the breadth of topics, including entropy and optimization, decreases the depth of coverage of more traditional probabilistic and statistical concepts to the point where students will find the concepts somewhat difficult to follow.

Chapter one, entitled “Uncertainty and decision-making,” provides a very nice overview of the role of probability analysis in decisions, although the use of decision trees is a little extensive. Chapter two, “The concept of probability,” is a comprehensive treatment of random events, with a somewhat heavy emphasis, again, on the use of decision trees and the definition of fair bets. Boolean algebra, Venn diagrams, and set theory are all included, albeit briefly. There is extensive coverage of conditional probability and Bayes’ theorem, and a reasonably balanced use of urn models to explain concepts. There is extensive use of graphical traces of repeated experiments and combinatorial analysis, which are actually a little hard to follow, and an unnecessary introduction of Bose-Einstein statistics, but a very nice example of the chance of the Titanic hitting an iceberg.

The third chapter, “Probability distributions, expectation and prevision,” is the single chapter devoted to this very important material. It covers all that it promises but at a fast pace (probability density and mass functions, cumulative functions, Bernoulli, binomial, negative binomial, geometric, hypergeometric, Poisson, and exponential and normal distributions) and even has space left over for multivariate distributions, coherence functions, conditional distributions, means, central moments, correlation, linear functions, and random processes (including Markov chains and autocorrelation functions).

With probability theory well in hand in the first 150 pages, the book devotes the following 50 to the concepts of utility theory. The approach is first one of economic theory, incorporating diminishing utility and risk aversion, followed by the more formal 0-1 utility of von Neumann and Morgenstern (and referenced to Raiffa’s 1968 book, Decision Analysis: Introductory Lectures on Choices Under Uncertainty, Addison-Wesley). Risk premiums and the concepts of maximum expected utility are discussed, as are multiple attributes and Bayesian analysis for preposterior analysis (expected value of perfect information).

The first half of the book is completed by one chapter on gaming theory and optimization (including Lagrange multipliers, the simplex algorithm, and duality) and one on entropy that includes a nice explanation of why the entropy associated with a discrete random variable is related to the sum of $p_i\ln p_i$ , leading to the exponential distribution as the maximum-entropy distribution when only the mean is known and the normal distribution when the mean and variance are known. The discussion on blackbodies and the Bose-Einstein distribution is less satisfying.

The book returns to probabilistic development with a chapter that includes a very nice development on single variate and multiple variate transformation of distributions (derived distributions), with special application to the multivariate normal distribution. The concept of convolution is introduced as a motivation for transforms that leads to sections on moment generating functions and characteristic functions. A rather intense introduction to random processes and power spectrum analysis is followed by illustrations of the central limit theorem (a somewhat confusing graphical approach, followed by a very nice development of the normal distribution as a limiting case of the binomial).

The eighth chapter continues with probability, focusing on the problem of uncertain parameters and moving quickly into Bayesian inference, including detailed examples based on the binomial distribution (and interesting comments on the historical justification for the use of a uniform prior). There is a very nice treatment of unknown mean and variance for the normal distribution, based on the use of conjugate priors, and a final example of the use of conjugate priors for the Poisson distribution. The remainder of the chapter returns to the more traditional problem of parameter estimation, using both parametric and nonparametric measures. Included is the derivation of confidence intervals for estimated means and variances and of hypothesis testing for assumed parameters, and a convenient table of confidence intervals for various distribution parameters.

Chapter nine is devoted to the subject of extreme values. There is a brief theoretical discussion of the extremum and order statistics of $n$ independent identically distributed variables, followed by examples with the exponential. Some sections are then given to discussion of return periods, characteristic values, and most probable maximum values. The presentation is clear and these are nice concepts not often included in an introductory book. This is followed by the general discussion of asymptotic extreme value distributions, with very little derivation but extensive examples and discussion. Following brief sections on weakest-link concepts and the Poisson process for rare occurrences, there is a nice but primarily qualitative discussion of Bayesian updating for extreme value distributions.

The book’s tenth chapter is a very nice overview of the concepts of risk, safety, and reliability. It begins with a general discussion of risk, including statistics on risk levels in society and a little explanation of risk perception. This is followed by a nice explanation of probability of safety with a single demand and a single capacity measure, and a brief discussion of series and parallel systems. Event trees are mentioned as natural consequences of the decision trees that are used throughout the book, and a fault tree is explained through an example, followed by some discussion of acceptable risk levels. There is a brief section on structural system reliability, but it is limited to early single demand and capacity concepts of second moment reliability from Cornell and only mentions invariance-based reliability by reference.

The final chapter of the book (other than a two-page philosophical closing) is devoted primarily to simple statistical techniques for analyzing observed data. These include computing the sample mean and variance as well as skewness and kurtosis, and the fitting of histograms, including the chi-square goodness-of-fit test. There is a nice derivation of least squares linear regression, including variability and confidence intervals for the computed slope and intercept. The chapter concludes with an introduction to Monte Carlo simulation.

The book contains three appendices. The first gives equations for 28 common distributions; the second contains brief discussions of a few mathematical functions; and the third contains solutions to selected exercises.