Principles of Data Analysis , Prasenjit Saha Cappella Archive , Great Malvern, UK, 2003. $12.00 paper (109 pp.). ISBN 1-902918-11-8

Available free online at http://ankh-morpork.maths.qmw.ac.uk/~saha/book

Probabilities are ubiquitous in physics. Physicists use them routinely in statistical and quantum physics, analyses of experimental data, and elsewhere. Encounters with probabilities typically involve abstract ensembles of identical systems or repeatable measurements, with probabilities interpreted as frequency ratios. This “frequentist” approach is the only way many physicists understand probabilities.

More generally, even if frequency data are unavailable, one may regard a probability as a likelihood based on prior information. The Bayesian approach, which goes back more than 200 years to Thomas Bayes and Pierre-Simon Laplace, enables one to consider probabilities without introducing frequency ratios.

In *Principles of Data Analysis *, theoretical physicist Prasenjit Saha focuses on Bayesian statistics and the maximum-entropy approach, a framework in which one identifies probabilities consistent with prior information such as average values and maximizes the entropy function to obtain best values for the probabilities. To get the Bayesian flavor, consider a data set *D*, a set of specified parameters ω, and a mathematical model *M*. If *P*(ω|*D,M*) is the probability of ω given *D* and *M*, then Bayes’s theorem is *P*(ω|*D,M*) = *P*(*D*|ω,*M*) *P*(ω|*M*)/*P*(*D|M*). *P*(ω|*M*) is the *prior* probability of ω, given the model *M* without any data. *P*(*D|M*) is constant for given *D* and *M* and *P*(ω|*D,M*) is a *posterior* probability that accounts for information contained in *D*. Specifying a useful prior *P*(ω|*M*) is a challenging aspect of the Bayesian approach. When Bayesian methods are applied to a given data set and two candidate models, they allow one to evaluate which model is favored by the data.

Saha’s coverage includes Bayes’s theorem; the binomial and Poisson distributions, with an example showing effects of choosing different priors; Gaussian distributions; the central limit theorem; random walks; the Monte Carlo technique (without explicit use of Bayesian concepts); least squares and distribution function fitting, both within a Bayesian context; information entropy; the maximum-entropy principle; and entropy in thermodynamics. The chapter on entropy and thermodynamics provides a succinct and clear, though relatively abstract, exposition of classical equilibrium thermodynamics and some statistical mechanics. Based on the maximum-entropy technique, the chapter does not require the material on Bayesian statistics that constitutes much of the book.

Saha’s writing style, though spirited, is terse. His thrust is on presenting a potpourri of illustrative examples and problems. The problems are graded according to difficulty, and Saha gives hints or answers for them.

Terseness can be a strength that enables a reader to glean the essence of the Bayesian approach relatively quickly. The price is that many details are omitted, and the reader is forced to fill in many steps or seek other sources. To learn the fundamentals, Saha recommends Edwin T. Jaynes’s * Probability Theory: The Logic of Science* (Cambridge U. Press, 2003; see also http://bayes.wustl.edu). From about 1958 until his death in 1998, physicist Jaynes wrote articulately about Bayesian statistics. He also did important work using the maximum-entropy framework. Saha also recommends Devinderjit Singh Sivia’s * Data Analysis; A Bayesian Tutorial* (Oxford U. Press, 1996). I suggest three additional sources written by physicists: Giulio D’Agostini, *American Journal of Physics*, volume 67, page 1260, 1999; Robert Cousins, *American Journal of Physics*, volume 63, page 398, 1995; and Volker Dose, *Reports on Progress in Physics*, volume 66, page 1421, 2003.

Some of Saha’s presentation could be more clear. For example, figure 2.1 displays computer-generated graphs illustrating the results of virtually flipping a biased coin. But the text does not specify either the degree of bias or the simulated data, namely, the number of heads. The mathematical level fluctuates from introductory to reasonably sophisticated and some of the manipulations lack sufficient motivation.

The book misses opportunities to link mathematics and physics. For example, Saha introduces the principle of indifference without mentioning its connection with the principle of *equal a priori* probabilities in statistical mechanics. Many of the 18 examples and 30 problems are not directly related to physics, and citations to the substantial literature on Bayesian methods in physics are lacking.

I commend Saha for making his book available by free download and minimal-cost paper copy. Despite some wrinkles, it provides a fresh, succinct view of data analysis at a level suitable for working physicists, graduate students, and very advanced undergraduates. In combination with the suggested supplements, this book could well serve as a stimulus and springboard for in-depth study and practical application of Bayesian and maximum-entropy techniques.