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\textbf{Bayesian Statistics: An Introduction} (3rd ed.) by Peter M.\ Lee. 
London:  Arnold Publishers, 2004. 351  + xv  pp. \$17.95. ISBN
0340814055.  

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Bayesian statistics has made great strides in recent years, due partly
to better  understanding of  priors  (e.g., automatic or reference
priors that can  be used in  the absence of subjective prior
information), partly due to the introduction  of  Markov Chain Monte
Carlo (MCMC) techniques for drawing samples  from the  posterior
distribution even when the sample space is huge, and partly due to the 
power  of  hierarchical Bayes models for  describing  very  complex
problems in  a  natural way. Added to this, the result  of  a  Bayesian
analysis naturally provides what scientists really would like to
know, whereas the interpretation  of  the results  of  a  standard
frequentist analysis is often unnatural and  confusing,  especially to
working scientists, but  even  to many  of  those with statistical
training. No longer is the Bayesian approach regarded with suspicion or
disdain  by  most classically trained statisticians; rather, some  of 
the most cutting-edge  work in statistics is being done by statisticians
 of  all stripes using the Bayesian  point-of-view. This is a tribute to
the power of  modern Bayesian methods.  The present book is the third
update of  a book that has become something of  a standard in
introductory texts on the subject.  It is not much different from the 
second edition, the  new material being a chapter on hierarchical Bayes
and an  expansion  of  the chapter  on MCMC (which was added in the
second edition).  However, the basic approach and most of the topics
remain virtually unchanged  from the second edition.  

I personally learned a great deal from the earlier editions of this book
when I  was discovering the power of Bayesian methods in my own work.
The selection  of  topics is basic, including chapters on inference for
normally distributed data  and for data having other distributions 
(e.g., binomial, Poisson and other sorts),  hypothesis testing (an area
in which the numerical results can differ substantially  from those
provided  by  standard hypothesis tests, particularly  for  point null 
hypotheses), two-sample problems, and Bayesian results on correlation,
regression and analysis of variance. A chapter on ``other topics''
discusses some of the  salient  features  of  Bayesian analysis that 
differ from  standard statistical  discussions, such as the important
Likelihood Principle (which standard methods  often violate), the
Stopping Rule Principle (which under mild conditions insulates  Bayesian
methods from  the  problems that arise in standard hypothesis testing 
when  the experimenter  is  allowed to  stop  optionally and
conditionally  on  the  results so far) and the role of decision theory.
The chapter on hierarchical Bayes  discusses the surprising Stein 
``paradox,'' whereby the obvious  estimator  for  a vector parameter under
square-error loss is inadmissible in classical decision  theory; better
estimators can be found naturally using a hierarchical Bayes model. 
Finally, a chapter  on MCMC  rounds out  the  book; MCMC  has 
revolutionized  Bayesian statistics over the past fifteen years by
providing a practical method for  obtaining results, especially
integrals,  by  posterior simulation. (In spirit these
interpretation  is  different: Standard statistics regards parameters as
fixed and  data  as  random variables, and averages or simulates over
the data, whereas  Bayesian statistics regards parameters as random
variables and the observed data  as fixed, and averages or simulates
over the parameters.) The chapter on MCMC  introduces most  of  the
standard techniques (with slice sampling  as a  notable  exception)
using examples coded in the free computer language R.  

The book has much to recommend it, but I do have some problems with it.
For  one thing, much  of  the  book  is devoted to exact or asymptotic
results, often  using so-called conjugate priors that with standard
error distributions produce  posterior distributions that remain in  the
conjugate family and are analytically  tractable. For example, using a
normal prior on the location variable in  normal  inference  problems 
with  a  known  data  variance results in  a  normal posterior 
distribution, so results are easily calculated. But, for all the
importance  of  such analytical techniques, they are too restrictive for
the vast majority of real-world  problems, which generally require
posterior simulation using MCMC to get  practical results. A student
reading this book might get an exaggerated idea of  the role that these
analytical techniques  play  in practical problems and might  regard
MCMC techniques  as  an  afterthought, whereas the truth is quite the 
opposite. It is for this reason that I no longer use it as a textbook in
my Bayesian  course, which  I  teach using simulation  as  the main
calculational technique,  introducing exact analytical results after
discussing problems  from a  simulation point of  view. My aim has
been to prepare the students to tackle real-world  problems in their
chosen field after this one-semester graduate course. The post-course
experience  of  my  students attests to the success  of this approach. 

Thus, I am of  two minds when recommending this book. Certainly I
learned  much from the first  edition,  so  it  can  be  useful  for 
self-study  by  a  mature scientist who is aware of its limitations; but
I would be careful about using it as  a textbook in a course, at least
without balancing it with other material (e.g., the  highly-regarded but
more advanced  book by  Gelman, Carlin, Stem and Rubin)  or with
lectures that placed more emphasis on simulation methods. 

\begin{flushright}
\textit{WILLIAM H.\ JEFFERYS \\
Harlan J.\ Smith Centennial Professor in Astronomy, Emeritus \\
Department of  Astronomy \\
1 University Station, C1400 \\
University  of  Texas at Austin \\ 
Austin, TX 78712-0259}
\end{flushright}

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\textit{Journal of Scientific Exploration}
\textbf{19} (1) (2005), 131--133.

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