Computational Bayesian statistics: An introduction. M. Antónia AmaralTurkman, Carlos DanielPaulino, and PeterMüller (2019). Cambridge, UK: Cambridge University Press. 243 pages, ISBN: 978‐1‐108‐70374‐1.

The challenge of mapping the ever-shifting territory of computational Bayesian methods is a daunting one: with the current torrential development of new methods and software in both the statistics and machine learning communities, any truly comprehensive treatment of the subject would necessarily be a bloated, constantly mutating document, nearly impenetrable to the incidental reader. One is reminded of the monstrous tome foisted on the narrator of Borges' (1979) The Book of Sand by a fey Orcadian Bible salesman: it contains an infinite number of books, revealing a different, confusing melange of text and images with each consultation; is strewn with inconsistent page numbers occasionally raised to the ninth power, and “is called the ‘Book of Sand’ because neither the book nor sand possess a beginning or an end.” The mere possession of the work eventually drives the narrator to the boundaries of his sanity until he voluntarily misplaces it in the basement of the National Library in Buenos Aires. Confronted with such a precedent, the authors of Computational Bayesian Statistics very wisely draw a line in the sand around the software and methodology associated with more traditional Bayesian statistical inference. The slender volume swiftly establishes Bayesian fundamentals, covers most of the more established and time-proven inference methods, and eventually concludes with its unique selling point: a comprehensive treatment of various software packages, chiefly BUGS, JAGS, STAN, BayesX, and R-INLA. The book offers a brief but coherent introduction to Bayesian theory, which mostly serves to establish notation for later in the book. It is unlikely to convert a dogmatic frequentist or offer much fundamental insight to those interested in the philosophical underpinnings of inference, but it does basically provide the material necessary for later discussion of Bayesian methods, with emphasis given to the subjectivist perspective. Objective Bayesian methods are given a short discussion, and some examples concerning Jeffreys’ prior are worked through in the univariate case. For the sake of completeness, one exercise indicates the failure of these methods to comply with the likelihood principle, while a footnote assures the reader that usually this is not a problem. After establishing the basics of Monte Carlo methods, the book returns to epistemological themes with a very welcome and neatly laid out chapter on Bayesian model assessment. Chapter 5 gives a clear overview of model validation, an often overlooked aspect of modern Bayesian methodology, through, for example, Bayesian p-values, different variants of information criteria and simulation-based strategies. The presentation is clear, appealing to intuition, while also providing references for the interested reader to follow up on. The inference chapters describe the basic topics fluently and sufficiently, with Gibbs sampling, Hamiltonian Monte Carlo, particle filters and reversible jump sampling forming the bulk of the Monte Carlo chapters, and INLA and variational methods dominating the presentation of distributional inference. Each of the subjects inevitably emerges like the head of a hydra: any attempt to dispatch them will necessarily introduce further questions and invite further explanation. The authors negotiate the Herculean challenge relatively successfully, although some subjects might have profited from further dissection before editorial cauterization. For example, the entire paradigm of likelihood-free inference is condensed into a guided exploration of elementary approximate Bayesian computation in a single exercise at the back of Chapter 6. Similarly, the treatment of sequential Monte Carlo methodology stops short of the important practical problems associated with resampling and weight degeneracy. The coverage of the material feels most restricted when it approaches more contemporary subjects: the hot topic of automatic differentiation for variational inference is given a perfunctory description within half a page. Perhaps the authors are unwilling to commit too much space on paper to a subject that may be obsolete by the next print edition, or they simply wish to maintain an appropriate scope for a book acting as an introductory text. The software chapter is perhaps the strongest asset of the book. There are thorough, well-worked examples of code and commentary, covering most popular Bayesian statistics packages. Maintaining online versions of the code examples in the book will be important, partly to ensure that they are consistent with contemporary versions of the software, but mostly to enable digital access for readers interested in executing the code themselves. There is no mention of software associated with the contemporary machine learning community, such as Tensorflow, Pytorch, or Julia, but more recent software packages are more likely to have less stable code syntax and user bases, leaving the validity of any worked code examples subject to the whims of future developers. Throughout the book, exercises serve the purpose of illuminating and expanding on ideas introduced in the preceding chapter. The themes of the problems follow the overall structure of the book: theoretical exercises, where, for example, the Bayesian crank is turned to produce a certain posterior from a likelihood and a prior, or implementation exercises, where the reader is asked to implement a certain algorithm for a specific model and data. A few of these exercises could benefit from some pseudocode or functioning R-scripts, as some implementation details are glossed over. Appendix B fills in some of these gaps by providing working code for a Gibbs sampler, but other algorithms could benefit as well. The exercises are frequently examples from the medical domain, and datasets are either provided by other packages in R or on the book's website. The book essentially provides a treatment of Bayesian methodology and software appropriate for a graduate-level course for mathematically literate students already familiar with the fundamentals of the Bayesian worldview. Such a situation is quite common within graduate statistics classes, which are often populated by arithmetically able students with undergraduate degrees in disciplines such as engineering, physics, or other fields of mathematics. The mathematical treatment is reasonably thorough to the extent that it is likely to intimidate applied scientists looking for an introduction to supplement their research: such students might be best guided toward existing applied texts such as Bayesian data analysis (Gelman et al. 2013). However, the book provides thorough references to academic statistics papers allowing students to examine the topics in greater depth, and many of the exercises serve as instructive tutorials for subjects that fall outside of the strict purview of the book. In this sense, the book acts as a powerful springboard for students to dive into the mighty deluge of Bayesian computational methods from our present-day position on the riverbank. It is inevitable that the course of the water will be flowing differently for future generations downstream, but perhaps Heraclitus provides the wisest perspective on the situation when he reminds us that you can never step in the same river twice.