As its title suggests, this text intentionally focuses on a few fundamental Bayesian statistical models and key computational tools. By avoiding a more exhaustive coverage of Bayesian statistical and computational techniques readily found in other texts, the authors have successfully cultivated an understanding of the process of creating and implementing a practical Bayesian statistical model for data analysis from beginning to end or, perhaps better-phrased, from prior to posterior. The end result is an exceptionally unique and well integrated “hands-on” treatment of Bayesian statistical modeling and computational techniques that will whet the dedicated student’s appetite for more complex Bayesian modeling and data analysis.

The authors’ format is consistent throughout the chapters, leading the student through an analysis of one or more data sets from beginning to end. The beginning of each chapter presens the student with a road map—a sort of annotated table of contents—of chapter topics. (Actually, the road map is preceded by a figure and a short quotation from books by Scottish novelist Ian Rankin, most of which, as the authors acknowledge, usually have little to do with chapter topics.) Each chapter focuses on one or more data sets introduced early in the chapter and revisited throughout the text and the exercises, which are purposefully placed within the text instead of at the end. This sets the stage for the development and discussion of appropriate data models, priors, posterior derivations, and computational techniques. Emphasis is placed on discussion of different prior distributions, which is appropriate given the text’s applied nature. The exercises vary in their level of difficulty, perhaps comparable to that found in the text by Casella and Berger (2002). Successful completion of the exercises is often necessary for the reader to fully understand current or subsequent developments in the text. A wide range of exercise types is included. I was particularly impressed by the repeated emphasis on posterior propriety. The foundational topics of Bayes theorem, conjugacy, and Jeffreys’ and G-priors are introduced within the familiarity of normal models, along with brief coverage of hypothesis testing, confidence intervals, and Bayes factors. Other topics include linear models and generalized linear models—probit, logit, and log-linear—as well as more specialized introductions to capture–recapture experiments, mixture models, dynamic models, and models for image analysis.

The powerful tool of latent variables is a recurring theme. Starting with the simple-to-understand latent variable representation of the probit model, the authors illustrate basic interpretive and sampling advantages of latent variables. With the basic idea established, the student is prepared to appreciate their power in the Arnason–Schwarz capture–recapture model wherein, in one example, the location of a lizard is specified as a partially hidden Markov model. Hidden Markov models reappear in the chapter on dynamic models. Coverage of latent variables continues in several other applications, including mixture models (where they are used to indicate a unit’s unobserved mixture component), as the state variables in the state-space representation of dynamic models, and, in the form of auto-logistic priors, Ising and Potts models (in their application to image segmentation as models for the underlying “true” image). The authors use Gibbs and Metropolis–Hastings algorithms throughout, and are careful to tailor the details of each as they revisit these algorithms with new applications (e.g., the Arnason–Schwarz capture–recapture Gibbs, the Potts Metropolis–Hastings). These and other applications gradually build the students’ confidence for modifying basic off-the-shelf algorithms for exploring sampling efficiency. Reversible-jump Markov chain Monte Carlo (RJMCMC) sampling for variable dimension models is introduced through mixture models with unknown numbers of components, and RJMCMC is again unsheathed to handle unknown orders in autoregressive and moving average models. Finally, few students will want to miss simulated tempering by pumping, which is used to provide extra boosts of energy to move among the hills of the posterior in its application to mixture models.

Bayesian Core is more than a textbook; it is an entire course carefully crafted with the student in mind. It is accompanied by a no-nonsense website providing supplemental material for both students and instructors. Data sets and R code (R Language and Environment for Statistical Computing) are provided for all but the introductory chapter and are integral to the text and the exercises. Both students and instructors will welcome the 459-page set of chapter-by-chapter lecture note slides in pdf. Instructors will find all of the necessary material, including graphics and macros, to typeset their own personalized version of the notes using LaTeX. In less than 5 minutes, I was able to download and create dvi and pdf files, complete with hyperlinks and without a hitch, using a standard LaTeX installation from Fink on Darwin 8.9.1 (Mac OS X 10.4.9). The website also maintains the inevitable errata, and an exercise solution manual is available to instructors through the publisher.

I was left wanting only a few things, which could be considered icing on the cake. These include a list of distributions, perhaps with annotation or with references to their use in the text, a list of symbols or functions used in the text, and perhaps a list of sampling algorithms.

Students and instructors will undoubtedly find the book’s pace brisk but refreshing, interesting, and fun. As an instructor of Bayesian statistics courses, I was pleased to discover this ready- and well-made, self-contained introductory course for (primarily) graduate students in statistics and other quantitative disciplines I am seriously considering Bayesian Core for my next course in Bayesian statistics.