This book’s full title is almost an abstract: *The Theory that Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted down Russian Submarines & Emerged Triumphant from Two Centuries of Controversy.* It tells a fascinating and engaging tale of how thinking related to Bayes’ Theorem influenced many aspects of human life in the last two centuries.

It may sound odd to an MAA member that an entire book could be devoted to that theorem, which many of us encounter while teaching elementary statistics, probability, or finite math. As the theorem is stated in those contexts, it is a simple and uncontroversial combinatorial fact about a 2×2 table. This book is really about the controversies that have surrounded the use and interpretation of that theorem. These seem, in fact, to predate the theorem itself.

Bayes did not actually state the theorem that bears his name. He discussed a thought experiment in which he used an initial guess and a sequence of additional bits of information to make a better guess, a pattern that later became common after the theorem was formulated and applied. Bayes did not even publish his work. That was left to a colleague, who used Bayes’ work to argue for the existence of God by reasoning backwards from empirical observations, and to similarly argue for causality against the contrary position of Hume. It was Laplace who formulated the theorem algebraically and applied it widely to issues we would regard as mathematical or physical rather than metaphysical. He did not, however, save it from controversy.

Most of the controversies have involved adding a temporal element to the theorem so that some probabilities become “prior” and some “posterior.” Many of the controversies revolve around whether the “prior” can be a guess or subjective opinion. Also at issue is whether it is reasonable to start by guessing all outcomes are equally likely.

Perhaps the most frustrating thing about reading this book was that the author often fails to make clear whether the many things called “Bayes” are the theorem itself or one of the many controversies surrounding the theorem. As a corollary, we often do not know which of these was crucial to the various applications discussed. There are exceptions to this, so the author appears to be aware of the issue, but perhaps considers it unimportant or too technical for a book for a general reader. This reader found it a bit like a biographical play in which the subject is named repeatedly but never comes on stage. An appendix does finally state the theorem in simple form and gives a worked example that gives little clue as to why it became controversial. Of course, a non-technical reader may not need or want to know any details about Bayes’ theorem. Still, most books on mathematics or statistics for a general audience have managed to give their readers a much stronger sense of their subject.

General readers may be quite amazed to learn how large a role mathematics and statistics have played in major events of our times. Much of the information the author has presented will be new to technical readers as well. That information is presented mostly in lively narrative that holds ones interest for the story rather than the technicalities. The author is at her best in extended stories where we get some glimpse of what aspect of “Bayes” was involved and how it solved a problem not readily solved otherwise.

Among the details are some minor technical errors counterbalanced by a glossary and extensive footnotes and references. The design, binding, and freedom from the errors spell-checkers do not find are exemplary. The material seems very well researched and generally well-written. The writer is a gifted story-teller who sometimes lapses into excessive vagueness.

Despite my misgivings, this is an interesting read. I recommend it for general readers and mathematicians content with a good story and not hoping to learn more about Bayesian statistics per se.

After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at statistics.com and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.