Bayesian Data Analysis, 2nd edn

Abstract

This wide-ranging book on statistical analysis would serve well as a text-book for self-study, for use in courses introducing Bayesian statistics, or for exploring more advanced model building. At the same time it is sufficiently comprehensive to be an excellent reference for someone who needs an overview of many common statistical models and how they can be analysed. It contains 22 chapters, divided into four parts: ‘Fundamentals of Bayesian inference’, ‘Fundamentals of Bayesian data analysis’, ‘Advanced computation’ and ‘Regression models’. The first two parts introduce the basic ideas behind Bayesian statistics for some simple models that can all be handled analytically. One could argue that inference and data analysis are two sides of the same thing. However, the reason that data analysis is separate from inference is that it deals with the practicalities of data analysis, whereas the inference part deals more with the theory. In this context, the practicalities of data analysis include the construction of hierarchical models, model validation, accounting for some problems related to data collection (e.g. censoring, truncation and randomization) and general modelling advice. The part on advanced computation discusses methods for computing posterior distributions and includes analytical approximations, the EM algorithm and Markov chain Monte Carlo methods. The final part covers almost two-fifths of the book and is devoted to regression models in a very wide sense. It covers both linear and non-linear models as well as mixture models and models for missing data. It is concluded by a chapter on decision analysis where three examples are used to show how posterior distributions can be used for decision-making. In addition to the four main parts, three brief appendices cover summaries of various probability distributions, sketches of proofs of asymptotic theorems, an introduction to Bayesian modelling using the software packages R and BUGS. It is unclear to me why the sketched proofs are included: this appendix is only four pages and could easily have been incorporated in the main text or omitted entirely. The emphasis of the book is on practical aspects of statistical data analysis and there are only a few theoretical derivations. The applied focus is achiev-ed by the inclusion of many well-chosen examples and analyses of real data. Several examples are used in more than one context, making it easy to understand how simple models can be developed into more sophisticated and improved models. The coverage of R and BUGS contributes to making the book very practical and this subject really deserves to be treated in a chapter of its own rather than in an appendix. The book is clearly written by people who believe firmly that statistical inference should be performed in a Bayesian framework. Comparisons between Bayesian statistics and classical statistics are made where appropriate and the book is generally balanced in its views. However, at places the authors have fallen for the temptation to promote the Bayesian point of view by bashing off classical statistics by using rather thin arguments. This problem is especially pronounced in Chapter 8 where Bayesian methods are compared with a somewhat simplified view of classical statistics. For example, the authors present both maximum likelihood estimation and unbiasedness as paradigms from classical statistics that lead to non-sensible results in some cases. On the basis of large sample arguments, these concepts are indeed very important in classical statistics, but that does not mean that non-Bayesians are completely unaware of the problems that can occur in small samples. Such biased compar-isons are unfortunate, as they may discourage people with a non-Bayesian background from reading the book and thus from benefiting from the discussions which are relevant to both views. For example, much of the discussion of model construction, validation and computation is highly relevant to non-Bayesian analysis of doubly stochastic models such as variance components, frailty models and Cox processes. Despite the occasionally biased comparisons with classical statistics, I enjoyed reading this book because it touches on such a wide range of statistical topics. Moreover, it largely equips the reader to perform Bayesian analyses without having to consult other books. The part on advanced computation is probably the only part that stops short of giving the reader enough understanding to apply it to analyses outside the examples of the book. However, people who want to use more sophisticated models or who would like a deeper understanding of certain topics will appreciate that the book provides historical background and plenty of references in the notes to each chapter. In conclusion, I would recommend this book to anyone who is interested in practical data analysis or who needs a reference book on Bayesian statistics and to non-Bayesian statisticians who want to broaden their ho