The EM Algorithm and its Extensions.

Geoffrey J. McLACHLAN and Thriyambakam KRISHNAN. New York: Wiley, 1996. ISBN 0-4711-2348-7. xvii + 274 pp. $59.95.

McLachlan and Krishnan remark in their Foreword that “it is surprising that ... no book [on expectation-maximization (EM) has appeared.” This is perhaps not too remarkable due to the fact that the seminal paper by Dempster, Laird, and Rubin (1977) synthesized and energized applications and research in the area, and so no book length treatment was needed for many years. However, given the recent pace of development of EM theory, applications, and extensions, the appearance of this well-written, book-length treatment is now appropriate.

Who might this book benefit? It is clearly intended as expository (it contains no exercises), and the authors suggest that it will be useful to both practitioners and researchers of statistics and will be accessible to graduates in Statistics (Bachelors level). This is a perhaps an overly optimistic reach. The book does contain numerous examples that lead the way in showing how the EM algorithm is to be applied. However, much of the book remains out of the reach of anyone without at least some familiarity with the research literature.

In my opinion, the book’s strongest contribution is in relating the various modifications and improvements of EM and in indicating relationships to alternate techniques. These are especially hard to establish on one’s own, because it would require gathering together research in different journals, by different authors, in different notations. I especially enjoyed the chapters on “Standard Errors and Speeding Up Convergence” (Chap.

4) and “Extensions of the EM Algorithm” (Chap. 5). The references are quite up to date.

I would have liked to have seen a bit more advice for the practitioner: general strategies for applying and implementing EM, stopping strategies, advice as to when EM can be expected to be better or worse than competing techniques, and some practical advice on when it can be expected to find a global maximum and/or how to help it do so. Some techniques seem to have been enthusiastically endorsed, even though the practitioner may have trouble implementing them. For example, for many problems, the Meng and Rubin (1991) method of obtaining standard errors can be numerically unstable.

Being slightly critical, I did not like a few portions of the book. The bootstrap section didn’t really explain the bootstrap and didn’t show how the bootstrap can be used to special advantage in missing data problems. The section on Newton-type methods claims that EM is “a method for finding Newton-type methods claims that EM is “a method for finding zeros of a function” (it is really more a hill-climbing technique and often works to find the maximum likelihood estimator (MLE) even when it is not zero of the likelihood function). Self-consistency was discussed and “proven” for the MLE withiut ever being formally defined.

That said, I would recomment The EM algorithm and Its Extensions to researchers wanting to learn more about the EM algorithm and variants. Practitioners and more applications-oriented statisticans may want to persue the examples but may prefer to check it out of the library rather than own it.

Charles E. McCULLOCH Cornell University

REFERENCES

Dempster, A.P., Laird, N.M., and Rubin, D.B. (1977), “Maximum Likelihood From Incomplete Data via the EM Algorithm” (with discussion), Journal of the Royal Statistical Society Ser. B, 1–22.

Meng, X.L., and Rubin, D.B. (1991), “Using EM to Obtain Asymptotic Variance-Covariance Matrices fro Missing-Data Problems Using EM,” Journal of the Americal Statistical Association, 86, 899–909.