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Bayesian probability theory : applications in the physical sciences
von der Linden W., Dose V., von Toussaint U., Cambridge University Press, New York, NY, 2014. 672 pp.  Type: Book (978-1-107035-90-4)
Date Reviewed: Jan 12 2015

I did not know enough, a half century ago, to decide between being a frequentist or being a Bayesian at the time an application of Bayes’ theorem forced itself into my physics thesis. Of course, “as everyone knows,” that theorem is agnostic with respect to those two philosophies of probability, that is, it is valid in both --and the mechanics of the theorem’s application generally comprise a relatively straightforward computation.

Though the proverbial drop of a hat is sufficient encouragement for me to reminisce (as above), the following assertions from the present book’s preface--with which assertions I very much agree--are fundamental to the book’s high relevance and utility: “[P]robability theory is at the heart of any science[;] it represents--in the words of E. T. Jaynes--the ‘logic of science.’” And it is apparently still true, as it was for me 50 years ago, that “[s]trangely enough, most scientists never had a thorough education in probability theory.” Some inspired but uncharacteristic realism led me at the time to refrain from embarking on any exploration or elaboration in that physics thesis, of such “collateral” topics as the foundations of probability.

When the work of the above-mentioned E. T. Jaynes appeared in book form [1] some decades later, it was for me a welcome-home revelation and, I’m sure, a validation of the thoughts of many scientists with respect to their understanding and application of probability theory. Jaynes, along with other leading lights in Bayesian probability, are given their due within the book and among the 219 references.

The usual initial random sampling put me first on page 36, the Monty Hall [MH] problem, that problem causing a sensation if not (un)civil war, beginning in 1975. This book treats MH in a clear and masterly fashion, in less than one full page. (There are numerous articles, book chapters, and books on the subject, for example, [2].) MH also struck me as the quintessential Bayesian paradigm: the probability of winning--of choosing the door behind which is the prize--depends on one’s use of all relevant information. (The authors are requested to forgive my simplistic and vague characterization, regarding which they bear no responsibility.)

The other initial sample was chapter 18, “The Frequentist Approach.” Simply put, this is the clearest treatment of frequentist hypothesis-testing essentials that I’ve come across. “[In] the frequentist’s reasoning there is no such thing as a probability of or for a hypothesis, as the latter is not a random variable.” There seems, in this chapter and in Appendix B.1, to be a well-paved (if not royal) road to the Neyman-Pearson lemma, which yields the optimal region for rejection of a hypothesis along frequentist lines.

One could justifiably characterize the range of this excellent work as encyclopedic, as long as one connotation of that term, namely “superficial,” is here rendered null; the book has both depth and breadth, and is extremely well written and organized. Its 31 chapters are spread over six parts: “Introduction,” “Assigning Probabilities,” “Parameter Estimation,” “Testing Hypotheses,” “Real-World Applications,” and “Probabilistic Numerical Techniques.” Regarding Appendix A, “Mathematical Compendium,” it is appropriately advanced and modern--and does not bring the word “cookbook” to mind. The derivations and proofs comprising Appendix B, including those of chi-squared and Student’s t, are ones that I wish would have been my first experience with these ever-applicable statistics (and others in the standard repertoire).

But the most important strength of this work is its efficient and effective explanation of both Bayesian and frequentist thinking and methodology. The appropriately rich and lengthy “Real-World Applications” (Part 5) will, I dare say, solidify any reader’s knowledge of, and confidence in, Bayesian probability theory, in a setting that comprises the actual practice of good science. Two old acquaintances, if not friends, that I met here are the Kramers-Kronig relations between the real and imaginary parts of the response function of a linear passive system, and (separately) the probability (in the form of a “cross section”) of Plutonium fission upon neutron capture. One must, in the end, believe the data along with the error bars. Bayesian analysis, though often yielding similar results to those of the frequentist approach, seems a tour de force in supplying such confidence. Other readers will find their own old friends in Part 5.

In having read Jaynes [1], the outstanding book by Lee [3], and the equally outstanding book under review, I have wondered all along about advocating the removal of the adjective “Bayesian,” which these works well transcend. These three works, along with others [4], seem a maximal realization of James Clerk Maxwell’s prophetic, “The true logic of this world is in the calculus of probabilities.”

This great book should not be missed.

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Reviewer:  George Hacken Review #: CR143076 (1504-0277)
1) Jaynes, E. T.; Bretthorst, G. L. (Trans.) Probability theory: the logic of science. Cambridge University Press, New York, NY, 2003.
2) Moscovich, I. The Monty Hall problem & other puzzles. Sterling Pub., New York, NY, 2004.
3) Lee, P. M. Bayesian statistics: an introduction (4th ed.). Wiley, Hoboken, NJ, 2012.
4) Calvetti, D.; Somersalo, E. Introduction to Bayesian scientific computing. Springer, New York, NY, 2007.
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