Kendall's Advanced Theory of Statistics, volume 2B, Bayesian Inference, 2nd edn

The first edition of this book was published in 1994, with O'Hagan as the sole author, but it appears never to have been reviewed in any of the Royal Statistical Society's publications. This was a serious omission, for the book provided a fine account of Bayesian ideas a decade ago. The modern Bayesian paradigm is essentially a product of the latter half of the 20th century and provides a logical development of the whole of statistics, including routines for implementation. However, the paradigm never became accepted in the manner that its enthusiasts had hoped because, although it was theoretically satisfying, it was inadequate on the practical side. With hindsight, the reason for this deficiency was not that it did not provide a good description of reality but rather that the Bayesian method made heavy demands on computing. By contrast, the collection of miscellaneous devices, that had been developed within the frequentist framework, could more easily be employed with the simple calculators that were used before the 1960s. Now that substantial computing power is readily and cheaply available, the Bayesian paradigm can encompass the whole of the frequen-tist stage and, more importantly, can enter parts of inference that others cannot reach. The second edition is about 50% larger than the first, mainly, but not entirely, because it includes an account of the computing developments of the past decade that have made the results more widely available. The book is part of the Kendall series on ad-vanced theory and follows the style of other volumes in the series, including substantial exercises at the end of each chapter. The text also contains copious references to other volumes in the series, which is admirable if you have them on your shelves, but occasionally a minor irritant is that the presentation is not always as self-contained as it might be. Notice that the topic is inference. There is brief reference to decision-making and utility but the great strength of the paradigm, that it alone leads to coherent decision-making, is not extensively treated. Bayesian theory is like other branches of applied mathematics in that it starts from basic postulates and proceeds logically to develop theorems and methods. It is natural for any exposition to follow the same course but the authors proceed differ-ently, delaying the axioms until Chapter 4, providing instead substantial introductory material, an approach that has much to recommend it and is more in line with much statistical thought. Some connections between the ideas of the two main schools of statistics, frequentist and Bayesian, are explored, thereby easing the reading for an experienced statistician wanting to learn about the new ideas. A sensible criticism of the paradigm is that it lacks flexibility in that it provides a model that it rigidly adheres to. The authors admirably meet this objection by providing two good chapters on model comparison and on robustness and model criticism. Earlier there has been some unexpected, but welcome, material on understanding distributions, especially multivariate forms. It is easy today to leave the computer to manipulate and produce its answers without the scientists properly appreciating the situation, and the authors show considerable sensitivity to this difficulty, giving the book a value beyond that of the formal mathe-matics. The two chapters on computation and on Markov chain Monte Carlo methods well describe recent developments, though progress here is so fast that no text can be fully up to date. There is material on graphical models, that have already shown their promise in helping us to understand the behaviour with many variables. A personal disappointment was the failure to mention Pearl's ideas on causality, a notion lacking in most statistical thinking. There are chapters on the linear model, discrete data models, nonparametric models and others, all clearly treated from the Bayesian perspective but incorporating the valuable concepts of the frequency school where these carry over into the new thinking, as they often do. Although this is a book on theory, it ends with two case-studies that illustrate some of the ideas present in the text. There is an extensive set of references running to 30 pages. If I have a criticism it is that the authors do not shake off the false ideas of frequentist statistics sufficiently, failing to adhere strictly and entirely to the concepts of coherence. But, the more that I read of their work, the more I begin to feel that they were right in being more flexible because it means that statisticians will be more easily able to appreciate what is new and valuable in the Bayesian approach that works entirely within the calculus of probab-ility. This is a really important addition to the sta-tistical literature, providing a careful, gentle ap-proach to modern ideas that should find favour as a university text and as a reference book for all of us.