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\noindent{\LARGE\textbf{A Review of Six Introductory Texts on Bayesian 
  Methods}}

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\noindent\textbf{Michael Smithson}

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Department of Psychology, The Australian National University

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\noindent\textbf{michael.smithson@gmail.com}.
 
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\noindent{\Large\textbf{Introduction}}

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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.

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\noindent{\Large\textbf{A Comparative Assessment}}

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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.

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\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}

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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].

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\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}
}

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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.

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\noindent{\Large\textbf{Suitability for Self-Instruction and Teaching}}

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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.

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\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}

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\noindent
\textit{Journal of Educational and Behavioural Statistics} \textbf{35}
(3) (2010), 371--374.

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