\documentclass{article} \begin{document} \noindent{\LARGE\textbf{A Review of Six Introductory Texts on Bayesian Methods}} \medskip \noindent\textbf{Michael Smithson} \medskip Department of Psychology, The Australian National University \medskip \noindent\textbf{michael.smithson@gmail.com}. \bigskip \noindent{\Large\textbf{Introduction}} \medskip Since 1990, Bayesian statistical methods have undergone major advances, both in estimation techniques (especially Markov chain Monte Carlo [MCMC] methods) and in software implementations thereof. During the first decade of this century, several textbooks on Bayesian statistics have appeared, each purporting to introduce Bayesian statistics and the new techniques to students and practitioners. Is a purely or even primarily Bayesian statistical methods curriculum for social science students feasible? Four difficulties impede this prospect: \begin{enumerate} \item The bulk of the research literature rarely uses Bayesian methods. \item A solid grounding in frequentist methods is needed to be able to read this literature in an informed and critical manner. \item A large majority of the current generation of social scientists is unacquainted with Bayesian methods. \item A solid understanding of Bayesian methods requires mathematical training that includes at least some calculus. It is unlikely that a majority of undergraduate social science students will come to introductory statistics classes with that background. \end{enumerate} None of these obstacles are insurmountable or fatal to the enterprise but clearly it will take some time before we see a Bayesian-dominated statistical methods curriculum in any of the social sciences. Graduate-level instruction clearly is the place to start and that is what the six texts reviewed here are intended for. Lee (2004) is the third edition of a relatively accessible text, whose intended audience was ``students who were reasonably well prepared mathematically and already knew some statistics'' (p. xv) but presumably not sufficiently prepared to profit from, for example, Gelman, Carlin, Stern, and Rubin (2004). Bolstad (2007) similarly states that his goal is to ``introduce Bayesian statistics at the earliest possible stage to students with a reasonable mathematical background'' (p. xiii) but has adapted his second edition to cater for those taking up his book at a more intermediate level. Marin and Robert (2007) claim their text is suitable for a 13- to 15-week course for ``scientists in all fields'' (p. vii). Ntzoufras (2009) presents his book as emphasizing model building and implementation using WinBUGS, possibly as the main or accompanying text in an introductory Bayesian course or as a self-taught tutorial. Gill (2008) and Lynch (2007) explicitly orient their books toward social scientists. Gill's stated purpose in the introduction to his 2002 edition is to provide a ``Bayesian methods book tailored to the interests of the social and behavioural sciences'' (Gill, 2008, p. xxix). Lynch claims that his text is more introductory than Gill’s, constituting a ``truly introductory'' treatment of Bayesian analysis ``involving typical social science models applied to typical social science data'' (p. vii). Thus, the authors of these six texts have somewhat different purposes and audiences in mind, but this review evaluates their suitability for instructing social science graduate students. \medskip \noindent{\Large\textbf{A Comparative Assessment}} \medskip All of these books require some grounding in statistics and introductory calculus, though they vary in how much background is needed. Table 1 summarizes these requirements, showing that Lynch and Bolstad demand the least, whereas Marin and Robert, Ntzoufras, and Gill require the most. Lynch and Bolstad provide brief coverages of their requirements. \medskip \noindent TABLE 1 \\ \textit{Required background} \\ \begin{tabular}{lll} \hline Book & Statistics & Mathematics \\ \hline Lynch & Minimal & Some calculus, linear algebra, covered \\ Bolstad & Minimal & Some calculus, covered \\ Marin \& Robert & Substantial & Calculus, linear algebra \\ Gill & Substantial & Calculus, linear algebra \\ Lee & Moderate & Calculus \\ Ntzoufras & Substantial & Calculus, linear algebra \\ \hline \end{tabular} \medskip Getting to the website for each book is straightforward except for Gill and Marin and Robert. The URL listed on page xxi of Gill’s book does not work. Instead, there are two relevant webpages http://jgill.wustl.edu/BMSBSA2 and \\http://jgill.wustl.edu/BMSBSA/index.html. The first link leads to Gill’s dedicated R package and the second to the data and code files for the first edition. Marin and Robert are coy about the location of their website. When writing this review, it could be found at http://www.ceremade.dauphine.fr/verb+$\sim$+xian/BCS/. Each book’s website has example data sets and code, where applicable, and an errata document. Marin and Robert also provide course slides. Lynch’s errata list is incomplete, but there do not seem to be a large number of errors. Gill still has quite a few errors, although not as many as plagued the first edition. All six books have exercises. Lee provides answers to all of his, Bolstad answers to selected exercises, and Lynch answers to the exercises in Chapters 2--5 only. All of them make use of R, but only Marin and Robert provide instruction on using it and Bolstad instructs in the use of his R packages. Gill, Lee, Lynch, and Ntzoufras make use of WinBUGS, with the first three providing introductory instruction therein and Ntzoufras (of course) being comprehensive on this topic. Table 2 presents a comparative summary of topics covered by the books, with ``T'' denoting thorough coverage and ``S'' indicating a sketchy treatment. Clearly, Gill is the most comprehensive (and the longest, exceeding the others by more than 200 pages). Lee and Lynch perhaps provide the greatest breadth with economy (Lee has 351 pages, Lynch 357) [in the top line N denotes Ntzoufras]. \medskip \noindent TABLE 2 \\ \textit{Topics Covered} \\ \hspace{-0.5cm} {\footnotesize \begin{tabular}{lllllll} \hline & & &Martin & & & \\ Topic & Lynch & Bolstad & \& Robert & Gill & Lee & N \\ \hline Bayes' theorem without probability &T&T& &T&T&S \\ Beta-binomial model &T&T& &T&T&S \\ Normal model, mean unknown &T&T& &T&T&S \\ Normal model, variance known &T&T& &T&T& \\ Normal, mean and variance unknown &T& & &T&T&T \\ Multivariate normal model &S& &T&T& &S \\ Other distribution models &S&S& &T&S&T \\ Priors &S&T&T&T&T&T \\ Hypothesis testing &S&T& &T&T& \\ Credible intervals &S&T&S&S&S& \\ Bayes factors &S&S& &T&S&T \\ Comparsion with frequentist methods &S&S&S&S&T& \\ Gibbs sampling &T&S&T&T&S&T \\ Metropolis-Hastings sampling &T&S&S&T&S&T \\ Additional MCMC topics & & & &T&S&S \\ Model comparison &S&S& &T& &T \\ Model checking and evaluation &T& & &T& &T \\ Linear regression &T&T& &T&T&T \\ General linear model &T& & &T& &T \\ Hierachical models &S& & &T&S&T \\ Decision theory & & & &S&S& \\ \hline \end{tabular} } \medskip As should be expected, Gill and Lynch provide examples oriented toward social science students, mainly limited to political science and sociology. Bolstad, Lee, and Marin and Robert do not. Ntzoufras provides several amid examples from other domains. Social science instructors using any of the latter four texts would need to supply examples demonstrating how a Bayesian approach can be applied in their discipline. Instructors from disciplines such as psychology and anthropology would have to construct discipline-relevant examples, regardless of which text they used. Lynch and, to a lesser extent, Gill also stand out from the other four in dealing with fairly large data sets, the computational requirements of which still are an issue in MCMC modeling. \medskip \noindent{\Large\textbf{Suitability for Self-Instruction and Teaching}} \medskip Which of the six books reviewed here is best suited for which purpose? Autodidacts with the requisite background in calculus, statistics, and linear algebra probably would get the greatest benefit out of Gill (breadth of relevant topics and in-depth coverage of MCMC issues) or Ntzoufras (thorough instruction in WinBUGS and greater variety of commonly used models). What about graduate-level teaching? For political science or sociology students, the choice seems to boil down to Lynch (more accessible) versus Gill (more in-depth on some topics). The choice is not as clear for students in other disciplines such as human geography, psychology, or anthropology. Instructors could use Lynch or Gill and supply their own examples from their discipline. But as long as they must do that, a worthy alternative is a combination of either Bolstad or Lee with Ntzoufras. The advantages therein are accessible treatments of theory combined with a thorough introduction to modeling in WinBUGS. Either way, it would seem that there remain some untapped markets for introductory Bayesian texts tailored to disciplines in the behavioral and social sciences. \medskip \noindent{\Large\textbf{References}} \begin{itemize} \item Bolstad W. M. (2007). Introduction to Bayesian statistics. 2nd ed.\ Hoboken, NJ: Wiley. \item Gelman A., Carlin J. B., Stern H. S., Rubin D. B. (2004). Bayesian data analysis. 2nd ed. London, England: Chapman and Hall/CRC. \item Gill J. (2008). Bayesian methods: A social and behavioral sciences approach. 2nd ed. London, England: Chapman and Hall/CRC. \item Lee P. M. (2004). Bayesian statistics: An introduction. 3rd ed. London, England: Hodder Arnold. \item Lynch S. M. (2007). Introduction to applied Bayesian statistics and estimation for social scientists. New York, NY: Springer. \item Marin J.-M., Robert C. P. (2007). Bayesian core: A practical approach to computational Bayesian statistics. New York, NY: Springer. \item Ntzoufras I. (2009). Bayesian modeling using WinBUGS. Hoboken, NJ: Wiley. \end{itemize} \bigskip \noindent \textit{Journal of Educational and Behavioural Statistics} \textbf{35} (3) (2010), 371--374. \end{document}