McLACHLAN, G.J. and KRISHNAN, T. The EM Algorithm and Extensions. Wiley, New York, 1997. xvii + 274 pp. $64.95/ £50.00 ISBN 0-471-12358-7.

The EM algorithm has become one of the most widely-used statistical tools, and it now finds many applications beyond just the incomplete-data situations for which it was originally developed. The algorithm has been modified in various ways to improve its performance, and it has been generalized to address ever broader classes of problems. It is not surprising that an entire book has been devoted to the algorithm and its extensions.

The introductory chapter describes the types of problems to which the EM algorithm was originally applied. The examples provide a quick and easy introduction to the basic ideas. An interesting section in this chapter gives a history of the EM algorithm, tracing its origins back to 1886! In the statistical literature of the 1950’s and 1960’s there are several instances in which an EM algorithm is applied to a specific problem. The paper of Dempster, Laird, and Rubin (1977), which gave the algorithm its name, was the watershed paper. The historical section of Chapter 1 also summarizes some of the literature following the Dempster-Laird-Rubin paper. Several of those papers identified methods, such as iteratively reweighted least squares and Henderson’s method, as special cases of EM.

Chapter 2 continues the introduction by describing a wide range of examples; Chapter 3 provides some basic theory, including convergence properties; Chapter 4 discusses modifications to speed up the convergence; and Chapter 5 describes extensions to the algorithm that allow its use in a variety of applications. Finally, Chapter 6 briefly covers a variety of Monte Carlo methods that ahve been incorporated into the basic EM algorithm. There is an appropriate emphasis on the various ways to accelerate the convergence, including Aitken acceleration, conjugate gradient methods, and hybrid methods incorporating modified Newton-Raphson steps. Methods of obtaining variance estimates are also considered in some detail. These methods include a bootstrap approach and the “supplemented EM” algorithm, one of two methods that carry the acronym “SEM” (the other is the “stochastic EM” algorithm).

Although there are no exercises, the book could usefully serve as a text, or more likely as a supplementary text, for a second or third course in applied statistics, or for a course in statistical computing. Its primary use, however, will be as a reference for applied statisticians or other research workers. The index is very good, and the examples allow easy access to the methods. The bibliography is extensive, serving both for attribution of contributions to the subject and for further reading.

The book is quite comprehensive in the topics addressed, although in many cases the reader is referred to the literature for the details. This is especially true for the variety of simulation methods addressed briefly in chapter 6, and it is only in this chapter that I felt the discussion was somewhat skimpy.

There are many texts and articles that discuss various aspects of the EM algorithm, but this is the only book to give a unified view, covering the basic methodology and the underlying theory. Although there is a sequential development, building ever more complexity into the algorithm, the numerous examples keep the discussion accessible, and provide interesting casual reading.

J.E. Gentle

Computational Sciences & Informatics George Mason University Fairfac, Virginia, U.S.A.