The Bayesian Choice: A Decision-Theoretic Motivation by Christian P. RobertReview by: Malay GhoshJournal of the American Statistical Association, Vol. 91, No. 433 (Mar., 1996), pp. 431-432Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2291425 .

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Book Reviews 431

But although readers will find these chapters entertaining, Gehan and Lemak unfortunately fail to take advantage of their considerable histori- cal research. They do a below-average job of context setting and relating historical developments to current practice and future issues in clinical trials. They fail to identify the scientific, technical, and ethical similarities and differences among agricultural, laboratory, and biomedical research and how these differences have energized developments. Transitions and background are sometimes inadequate, especially for the uninitiated (for example, see p. 22 on Bayesianism).

A more synthetic and less linear approach to the history with foreshad- owing of current practice and trends would have added some excitement and focus. In fact, the book would have been more successful had it started with a fairly complete description of one or two clinical trials that high- light scientific, ethical, and technical issues. Then, the history could map a route from "The Dawning of Statistics" (Chap. 1) to the present and then the future. Chapters 1-4 are history written by nonhistorians, and the lack of training and experience shows. (Of course, this review is written by a nonhistorian and may be similarly inexpert.)

The first four chapters consume 127 pages, leaving only about 60 for the last two (Chapter 5, "Clinical Trials in the United States," and Chapter 6, "Designs and Analyses of Studies of Historical Importance") and no pages for prognostication. In Chapter 5 the book takes an abrupt turn in both style and content, suggesting a change in authorship. The style is less interesting and the coverage that of convenience. Most (but to be fair, not all) of the examples come from cancer research. Though the cancer focus is understandable in the light of Gehan's experience and contributions, many innovations and issues have been spawned by research on other diseases as well. More balance (with the attendant additional research) would have been welcome.

With Gehan having played a central role with Freireich in promoting the historically controlled trial (HCT), it is no surprise that the authors give considerable space to controversies surrounding the choice between HCT and the randomized controlled trial (RCT). Much is made of the apparent successes of the HCT (see, for example p. 144 on MOPP therapy), but the countervailing evidence is downplayed. A more balanced approach (pos- sibly imbalanced in favor of randomized studies!) would better represent the respective roles of these designs in acquiring scientific information.

Pages 146-149 attempt to make the case against randomized studies, quoting several critics of the RCT and its ethical underpinnings. The au- thors appeal for support from HIV/AIDS activists and misread the current situation somewhat. Gehan and Lemak accurately report that patient ac- tivism is a modern development of considerable importance. In HIV/AIDS research, activism has led to important changes in the organizational struc- ture and science of clinical trials, as well as considerable changes in the FDA's evaluation of emerging therapies. Although most AIDS activists in the late 1980s argued against the RCT, in the last several years many have lobbied for this design. Statisticians played a key role in communicating with activists, involving them in planning and evaluating trials and thereby developing advocates of the RCT.

My focus on the RCT/HCT controversy reflects the play it gets in the book and its continuing importance, especially in the context of the current excitement generated by outcomes research. Proponents stress the appar- ent monetary, time-frame, and ethical efficiency of historically controlled and uncontrolled studies in contrast to the apparent expense and ethical problems of the RCT. Yes, RCTs can be made more efficient; Europe leads the way with its large, simple trials. But most of the expense of a properly scoped and designed RCT supports collection of high-quality, standardized data on all participants. Randomization in and of itself is very inexpen- sive. RCT's are ethical when there is clinical equipoise (uncertainty among treatment options) for patients and clinicians. HCT's and data bases do an excellent job of generating clinical questions, providing baseline data nec- essary for the design of a study, and broadening the inference from a RCT. But the RCT is outcomes research at its best.

Many interesting issues could have been included or given expanded historical development. For example, though Gehan and Lemak consider interim monitoring, they also should have discussed issues such as estima- tion after testing. Blinding gets little attention, especially in terms of how it can change the clinical question. Issues surrounding intent-to-treat anal- ysis and adjustments for compliance gets little play, though they are at the top of the current list of controversies and central to the evaluation of his- torically controlled studies and the use of data bases. Bayesian approaches deserve additional discussion, as does the role of covariate adjustment and identification of patient subgroups. Adaptive allocation to balance on prog-

nostic variables receives some attention. Discussion of the relatively rare use of adaptation on outcome variables to assign relatively more patients to the apparently superior treatment would have led to commentary on the ethics of altering a clinical trial design on the basis of trends that have not reached clinical or statistical significance.

Common analytic techniques are mentioned in an ineffective subsection of Chapter 6. The presentation has little to offer the nonstatistician and will not be news to the statistician. This space would have been better used to expand on current issues and future directions. Meta-analysis (research synthesis, overviews) is an important development with implications for the future. Gehan and Lemak sketch the technique but fail to mention its sociological and scientific implications. For example, data registries such as the Cochrane collaboration at Oxford are facilitating, standardizing, and otherwise improving clinical trial synthesis. Interestingly, on page 186 Gehan and Lemak catalog problems with meta-analysis, including noncomparability of treatments, differences in study designs, and varying patient populations. These criticisms, although relevant to meta-analysis, gain considerable force as criticisms of the HCT and use of data bases.

Successful clinical trials depend on technical statistical inputs and on broad statistical science. Statisticians and statistical centers play active roles in developing forms and procedures, quality assurance, adverse event and clinical event systems, and training. Outlining these roles would have led into a discussion of necessary organizational ingredients for a suc- cessful clinical trial and the necessary staffing and budget for a statistical center.

The references relate well to the text and provide a broad array of historical and contemporary information. The name index is excellent; the subject index only adequate.

In Angle of Repose, Wallace Stegner writes of the Doppler effect in history; how the approaching future increases in pitch, with the pitch low- ering as the future becomes the present and then recedes as the past. His- tory, with a lower pitch, can help frame the present and future. Stegner's metaphor encourages linking the present with past and future. Although Gehan and Lemak provide interesting historical snapshots and some hint at current issues and developments, they fail to uncover and evaluate many of these important connections and trends.

Thomas A. LOUIS University of Minnesota

The Bayesian Choice: A Decision-Theoretic Motivation.

Christian P. ROBERT. New York: Springer-Verlag, 1994. xiv + 436

pp. $49.

The book is a welcome addition to the literature on Bayesian analysis and decision theory. It contains a wealth of material, both classical and modern, in these two necessarily overlapping subject areas, and it is ideally suited as a text for a graduate-level course on Bayesian decision theory.

The pre-1985 development in this general area is admirably captured in Berger's now-classic text (Berger 1985); however, the growth in Bayesian methodology as well as in Bayesian computing over the past decade has been simply phenomenal. Robert has succeeded brilliantly in integrating many of these recent developments in his book. Chapters 8 and 9, dealing with hierarchical and empirical Bayes analysis and Bayesian calculations, are substantial chapters containing very lucid accounts of some of these developments. Another example is Section 3.4, which introduces reference priors very nicely.

The book contains 10 chapters:

1. Introduction 2. Decision-Theoretic Foundations and Statistical Inference 3. From Prior Information to Prior Distributions 4. Bayesian Point Estimation 5. Tests and Confidence Regions 6. Admissibility and Complete Classes 7. Invariance, Haar Measures, and Equivariant Estimators 8. Hierarchical and Empirical Bayes Extensions 9. Bayesian Calculations

10. A Defense of the Bayesian Choice.

The chapters are all well written and pay careful attention to the motiva- tion, technical details, examples, and exercises. The examples often bring out the historical perspective behind the growth of a certain topic and sometimes illustrate the controversies underlying the different principles of statistical inference. The exercises included at the end of each chapter not only provide a good review of the material included in the text but also

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432 Journal of the American Statistical Association, March 1996

supplement this material admirably. Some of my favorite sets of exercises are 3.21, dealing with the natural exponential family with quadratic vari- ance function, 3.47-3.53, dealing with the famous marginalization para- dox, and several new results on James-Stein estimation in Chapter 8.

There is a clear emphasis on decision theory, and topics such as admis- sibility and minimaxity are very adequately addressed. A central point in decision theory is the specification of a loss, and, not surprisingly, many examples and exercises in this book are loss oriented. This may be unap- pealing to many applied Bayesians, especially those holding the view that the posterior distribution summarizes information about the data and only descriptive measures of the posterior such as the means, medians, standard deviations, quantiles, and so on are of interest. Although I sympathize with this view to a certain extent, I nonetheless side with the author here. The main emphasis of this book is integration of Bayesian analysis with deci- sion theory, and this synthesis cannot be achieved without specifying the loss.

The book contains several semesters worth of material. The author sug- gests topics that should possibly be included in a one-semester course. This suggestion excludes Chapter 8. But my own opinion is that even a one-semester course should possibly include some discussion of hierar- chical and empirical Bayes analysis. Applied Bayesians, and possibly even some non-Bayesians, use these methods on a routine basis, and an initial exposure to the topic will prove beneficial for a large body of students, irrespective of their area of specialization.

Robert's book has very little overlap with the recent publication of Bernardo and Smith (1994). The latter contains a more detailed account of Bayesian history, a subjective view of probability, foundations, model- ing, remodeling, and inference, but greatly deemphasizes decision theory. Thus, the materials in these two books will usually supplement each other, and a Bayesian may want to keep both these books in his or her library for a quick access to topics that are otherwise scattered in numerous journals.

If the current growth in Bayesian methodology continues at the same or higher rate, then any textbook in the subject is bound to become obsolete in a decade or so. Perhaps, more immediately, in the next edition of the book (which hopefully will not be too far away), Robert will expand the sec- tion on Bayesian robustness, the previous section on noninformative priors (which should include probability matching priors and its variations), and possibly include topics such as Bayesian model diagnostics, model se- lection, and others. Even without these topics, The Bayesian Choice: A Decision-Theoretic Motivation will make a great text for Bayesian analy- sis, decision theory, or any combination thereof.

Malay GHOSH University of Florida

REFERENCES

Berger, J. 0. (1985), Statistical Decision Theory and Bayesian Analysis (2nd ed.), New York: Springer-Verlag.

Bernardo, J. M., and Smith, A. F. M. (1994), Bayesian Theory, New York: John Wiley.

The Inverse Gaussian Distribution: A Case Study in Exponential Families.

V. SESHADRI. New York: Oxford University Press, 1993. xi + 256 pp. $60.

This book is a thorough and authoritative review of the inverse Gaussian distribution, complete with mathematical details. It is brimming over with the author's enthusiasm for the two subjects that have kept him busy for at least 20 years: exponential families and the inverse Gaussian distribution. The exponential family theory provides more than just background for the study of the inverse Gaussian distribution, partly because the ramifications of inverse Gaussian distribution theory takes the author into many diverse fields of statistics and probability ranging from stochastic processes to generalized linear models. This book sets the inverse Gaussian distribution in a much broader context than does the work of Chhikara and Folks (1989). In comparison, the latter is a leisurely stroll through the gallery of properties of the inverse Gaussian distribution, whereas Seshadri's book will test the reader's knowledge and perseverance. This is a specialized topic, but if exponential families is your business or the inverse Gaussian distribution your fancy, this book belongs on your shelf.

The required background depends on how far you go. Anybody can enjoy the extensive historical survey of the inverse Gaussian distribution in Chapter 1, and much of the rest of the book is accessible to a persistent

graduate student with a thorough background in mathematical statistics and probability. But a strong background in probability theory is needed to fully appreciate Chapter 5, on inverse natural exponential families. In fact, as Seshadri notes in the preface, Chapter 5 was written by Gerard Letac "in Bourbaki style," which is noticeably different from the engaging and direct style of the remaining chapters. It is hard to imagine this book serving as a textbook, but each chapter ends with a set of exercises (although many exercises refer to published results or are full-fledged research problems). The book's main use will be as a researcher's reference, or possibly as a reference in an advanced graduate course on exponential families or distribution theory.

The six chapters deal with fairly distinct topics and can to some extent be read independently of each other. Following a historical survey in Chapter 1, Chapter 2 deals with the properties of the inverse Gaussian distribution derived in the context of exponential families. Chapter 3 takes the reader through several different techniques for characterization of the inverse Gaussian distribution, ranging from Khatri's characterization in terms of independence of the sample mean and the sum-of-reciprocals statistic to the use of random continued fractions. Chapter 4 deals with a number of related distributions, ranging from mixtures involving the inverse Gaussian distribution to combinations and multivariate distributions. Much of the material in this chapter is of fairly recent origin.

Chapter 5, on inverse natural exponential families, provides an explana- tion of M. C. K. Tweedie's idea of "inverse" statistical variates, related to the interpretation of the inverse Gaussian as the first-passage time distri- bution for Brownian motion with drift. This led Tweedie to propose the name "inverse Gaussian distribution." This chapter introduces the reader to, among other topics, natural exponential families with cubic variance functions, the Levy-Khinchine representation of infinitely divisible distri- butions, and the Tweedie class of natural exponential families, also called the Tweedie scale.

Finally, Chapter 6 summarizes the statistical properties of the inverse Gaussian distribution. It includes some technical results, such as the cal- culation of unbiased minimum variance estimators, as well as a brief dis- cussion of the use of the inverse Gaussian distribution as error distribution in generalized linear models. No data examples are considered. The book ends with an extensive list of references and author and subject indexes.

Few distributions other than the normal have ever been given such a royal treatment as the inverse Gaussian distribution gets in this book. Why does it merit such a detailed and extensive account? One answer comes from the parallels between the inverse Gaussian and the normal distri- butions, which prompted A. P. Dawid to ask, in the discussion of Folks and Chhikara's (1978) review of the inverse Gaussian distribution, "where do these F and x2 distributions come from?," in reference to some of the sampling distributions for the inverse Gaussian distribution. Around 1980, G. A. Whitmore made a conjecture about the decomposition of the inverse Gaussian deviance into independent chi-squared terms that, had it been true, would have meant to the inverse Gaussian distribution what Cochran's theorem means to the normal distribution. A counterexample to Whitmore's conjecture was given by Letac and Seshadri (1986). But as the results in Chapter 6 illustrate, the parallels between the inverse Gaussian and normal inferences are still striking.

The inverse Gaussian distribution has often been pivotal in the develop- ments of exponential family theory. For example, Barndorff-Nielsen (1978, p. 117) gave the inverse Gaussian distribution as the prime example of a steep but nonregular exponential family. Another case is Daniels' (1980) result that only the normal, gamma, and inverse Gaussian distributions have exact saddlepoint approximations. The inverse Gaussian distribution is a prime example of a natural exponential family with cubic variance function, which in turn led to the classification of all such families (Letac and Mora 1990). It is also a prominent member of the Tweedie class men- tioned earlier. That there are still interesting aspects of the inverse Gaus- sian distribution waiting to be explored may be illustrated by a recent result on convergence of exponential dispersion models to members of the Tweedie class (J0rgensen, Martinez, and Tsao 1994), which includes as a special case convergence to the inverse Gaussian distribution.

The book has a number of typographical errors, and no serious reader should be without the one-page list of errata available from the author. I found a few other minor problems, mainly in the indexes and exercises. On page 76 is a result regarding a conditional distribution of generalized in- verse Gaussian form attributed to "P. Vallois (1988)." But the name Vallois is nowhere to be found in either the reference list or any of the indexes. Seshadri does not mention that the result is a simple corollary of a more general result for the generalized inverse Gaussian distribution (J0rgensen

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