The EM Algorithm and Extensions, by Geoffrey J. McLachlan and Thriyambakam Krishnan, New York: Wiley, 1997, ISBN 0-471-12358-7, xvii +274 pp., $59.95.

Over the last 20 years, substantial effort has gone into understanding, developing, and extending the expectation-maximization logarithm, which is popularly known as the EM algorithm. Though this type of algorithm was in use before 1977, Dempster, Laird, and Rubin (1977) wrote a phenomenal article in which they gave a general formulation of this algorithm and the name EM algorithm. The number of works in this area is quite large, as evidenced by the bibliography of this book, and it continues to grow. This is one of the first books on this subject, and the authors have done an excellent job organizing and presenting the materials and providing an up-to-date bibliography of the research papers.

The monograph is organized in a suitable manner so that it is accessible and useful to statisticians interested in applications research. The writing style is clear and easy to understand. It should be comprehensible to graduates with statistics as their major subject. Throughout the book, the theory and methodology are illustrated with several examples, and analytical examples are followed up with numerical computations wherever relevant.

The book contains six chapters. Chapter 1 begins with a brief introduction and calculation of maximum likelihood estimation when a likelihood equation does not admit a closed-form solution. Then the heuristics of the EM algorithm are introduced through missing-data problems. It is followed by the formulation of the algorithm, and then the terminology and notations are established. Bayesian perspective is also included by showing how it can be used to compute maximum a posteriori (MAP) estimates. This chapter also presents a summary of the properties of the EM algorithm and finally closes with the history of it.

Chapter 2 is composed of many examples. These cover missing-value problems, the multinomial distribution as used in genetics, the multivariate t-digit n for the provision of a robust estimate of a location parameter, and so forth. It also discusses the Poisson regression models and fitting of normal mixture models.

In Chapter 3, theoretical aspects of the EM algorithm are presented. It discusses the monotonicity property and rates of convergence, along with the generalized EM algorithm ant its properties. The theoretical viewpoint is emphasized here with some examples, and hence readers interested in applications only may skip the chapter altogether without losing continuity.

Chapter 4 tries to defend the EM algorithm against two major criticisms, (1) possibility of slow convergence and (2) no provision of covariance matrix (or standard error in the univariate step). Although discussing the first issue, it describes several methods for computing the estimated covariance matrix of the maximum likelihood estimator that are typically based on the observed information matrix. To answer the second issue, it discusses some methods that have been proposed to accelerate the convergence problem. Among these, Aitken’s acceleration procedure, the generalized conjugate gradient approach, and the hybrid methods of Newton-type and EM algorithms are presented.

Chapter 5 discusses the further modifications and extensions of the EM algorithm. The natural extension of the EM algorithm, popularly known as the expectation-conditional maximum (ECM), is the major point of discussion. Other variations and extensions of the ECM algorithm are also covered in this chapter.

Chapter 6 contains a very brief discussion of other computational techniques like the Monto Carlo E step, the stochastic EM algorithm, data augmentation, and Gibbs sampling. The book is then concluded with some of its interesting applications in Markov models, AIDS epidemiology, and neutral networks.

Debashis Kushary Rutgers University