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Bayesian theory. (English) Zbl 0796.62002
Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley & Sons. xiv, 586 p. £60.00 (1994).

This textbook is the first volume of a related series of three volumes – Bayesian Theory, Bayesian Computation and Bayesian Methods – which are set to provide an up-to-date overview of the Why? How? and What? of Bayesian Statistics, as perceived by the authors. This volume focuses on the basic concepts and theory of Bayesian statistics, which cover elementary foundations, mathematical generalizations of foundations, modelling, inference and remodelling. Here, the primary emphasis is placed on general ideas. The material of the book is covered in nine chapters followed by two Appendices, References, Subject Index and Author Index.

In Chapter 2, the concept of rationality is explored in the context of representing beliefs or choosing actions in situations of uncertainty. The importance of a decision-oriented framework is stressed with the approach that statistical inference should be viewed simply as a particular form of decision problem. The dual concepts of probability and utility are formally defined and analyzed within the decision making context. In Chapter 3, the ideas and results of Chapter 2 are extended to more general mathematical setting.

The role of familiar mathematical forms of statistical models and the possible justifications, from a subjective perspective, is examined in Chapter 4. The key role of Bayes’ theorem in the updating of beliefs about observables in the light of new information is identified and related to conventional mechanisms of predictive and parametric inference, in Chapter 5. The prospect of systematically entertaining a range of possible model beliefs, rather than predicating all analysis on a single assumed model, is examined in Chapter 6.

Summaries of main univariate and multivariate distributions that appear in the text, are collected in Appendix A with the summaries of the prior/posterior and predictive forms. Classical decision theory, frequentist procedures, likelihood theory and fiducial and related theories are reviewed in Appendix B.

Reviewer: K.Alam (Clemson)

62A01Foundations and philosophical topics in statistics
62-02Research monographs (statistics)
62C10Bayesian problems; characterization of Bayes procedures
62C12Empirical decision procedures; empirical Bayes procedures
62-01Textbooks (statistics)