Bayesian Methods: An Analysis for Statisticians and Interdisciplinary Researchers
Albert, James H., Journal of the American Statistical Association
Thomas LEONARD and John S. J. HSU. New York: Cambridge University Press, 1999. ISBN 0-521-59417-0. xi + 333 pp. $64.95.
In this book's Preface, the authors comment that the state of statistical science is continuously evolving, and that it is important for applied researchers to be able to use the new methodology with specific knowledge of the assumptions involved in these methods. They describe the two schools of statistical thought, the "Fisherian" (usually called classical or frequentist) and Bayesian philosophies, and state that Bayesian methods have a number of advantages over the Fisherian procedures, including good long-run frequency properties. This book aims to show how Bayesian statistical methods can be used in drawing scientific, medical, and social conclusions from data. The book is intended for use by masters-level students learning statistics from both Fisherian and Bayesian viewpoints, by interdisciplinary researchers working in statistical modeling in their own area, and by doctoral students interested in research in Bayesian methodology. I was very interested in this book, as I see a need for texts that show the advantages of Bayesian methodology in a variety of applied statistical settings.
Chapter 1 sets the stage for the Bayesian material to follow. The first section defines the likelihood function, the maximum likelihood estimate, and different information criteria [such as the Akaike information criterion (AIC) and Bayesian information criterion (BIC)] that can be used to compare models. Large-sample properties of likelihood-based procedures are described, and a number of examples are worked out carefully to illustrate the generality of a likelihood approach to make inferences for a particular model, or to compare models. Chapter 1 concludes with statements of important results relative to likelihood inference, such as the factorization theorem, the Cramer-Rao and Blackwell-Rao theorems, the likelihood inference, and the use of a profile likelihood to eliminate nuisance parameters. It seems that this chapter would serve as a useful review for those students who have been exposed to likelihood inference, but that the first-year graduate student would find some of the later topics, such as a profile likelihood, hard to follow.
Bayes's rule in the discrete setting is the focus of Chapter 2. The use of Bayes's theorem in medical diagnosis and legal settings is described, and an example illustrates Bayesian inference for a discrete-valued parameter. Section 3 describes choosing between a discrete set of models, and Bayes's rule together with a Schwarz approximation to the integrated likelihood are shown to give simple expressions for the posterior model probabilities. This chapter concludes with an interesting example showing how Bayes's rule can be applied in logistic discrimination.
Chapter 3 describes the implementation of the Bayesian paradigm in the one-parameter case. This chapter covers assessment of a prior distribution, computation of the posterior and predictive distributions, and various ways of summarizing the posterior distribution. There is disagreement among Bayesians on how to construct a test of a point null hypothesis of the form [theta] = [[theta].sub.0]. The proposal here is to compute a "Bayesian significance probability" P([theta] [leq] [[theta].sub.0]\ data) to make a decision. The authors do not advocate placing a positive probability on the value [[theta].sub.0], which was discussed by Berger and Sellke (1987). Bayesian inference for many of the standard examples, such as binomial, Poisson, and normal mean with known example, are described here. Some of the examples, such as inference for an upper bound of a uniform distribution and normal mean inference using a uniform prior on a bounded interval, seem a bit artificial but may be helpful for practice in computing posterior and predictive densities. One section discusses the choice of suitable vague priors for a single parameter, including Jeffreys's invariance prior. …