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Review  The Theory that would not Die  Sharon Bertsch Mcgrayne

Occasionally I review a book that makes me think 'I wish I wrote that'  and sometimes I nearly did. The subject of Sharon Bertsch Mcgrayne's book, as the rather lengthy subtitle tells us is 'how Bayes' rule cracked the Enigma code, hunted down Russian submarines and emerged triumphant from two centuries of controversy.' There is no doubt that Bayes' theorem is the most intriguing piece of maths most people have never heard of, and I did once write a proposal for a book about it, but the publisher said no one would get it. I believe they should get it. But Bayes' theorem, though simple, is famously difficult to keep in mind. So a significant test of this book is how well Mcgrayne gets across what the theorem really is.
The good news is that this isn't a stuffy book of heavy mathematics  Mcgrayne has a light touch and an airy style. I did worry early on if it was too airy as she resorts to language that is a little cringeworthy. She says 'In 1731 [Bayes] wrote a pamphlet  a kind of blog'  now if she had said 'if he was alive today he would probably have written a blog' I would have been comfortable. But to put it the way she does... I can imagine her writing about Shakespeare: 'Around this time, Shakespeare wrote his first play  a kind of movie.'
This is mildly worrying, but what is more concerning is the way she handles the topic of another pamphlet Bayes wrote. It was, it seems, a response to George Berkeley's 'The Analyst: A discourse addressed to an infidel mathematician.' The infidel in question was Edmund Halley, an atheist, and concerned calculus. Berkeley's points out that Halley mocks believers for taking things on faith, yet supports a mathematical concept that requires you to do maths with something that disappears, as Berkeley puts it 'The ghosts of departed quantities', which also takes faith. In his quite detailed analysis, Berkeley points out a legitimate mathematical flaw in the basis of the calculus, as practised at the time.
But Mcgrayne's take is quite different. She calls it an 'inflammatory pamphlet attacking Dissenting mathematicians and... "infidel mathematicians" who believed that reason could illuminate any subject.' That is patently wrong. Halley was not a Dissenter in the usual sense of the word, and Berkeley's attack on the basis of calculus was, mathematically, correct. Berkeley was, in reality, arguing for the use of reason and at the same time attacking Halley's lack of Christian faith, something Bayes would have heartily agreed with. What worries me is if the reality of Berkeley's pamphlet could be so distorted to fit a particular viewpoint, how many other historical facts have been misused? This might be a single instance, but it was a bit worrying, coming as it does on page 4.
The bulk of the book concerns the 200 year battle between two types of statistics. Broadly there is frequentist statistics, the one you are likely to be familiar with, where you gather lots of data and spot trends, calculate means and all that good stuff. Then there is Bayesian statistics. This starts with an prior knowledge, or probabilities you might have, even if not directly about the problem in hand, then transforms this prior knowledge with new data as and when it is available. This means it can produce useful results with far less data  a more typical real world situation  but the maths can be quite messy, and it has a degree of subjectivity that mathematicians have always shied away from.
I did a masters in operational research in the 1970s, a discipline that Mcgrayne tells us was founded on Bayesian statistics, but never once heard anything about them on my course. This shows just how much fashions have often swung against Bayes.
So how does the book do? Not brilliantly. It is irritating vague about how Bayesian statistics works, combining a totally opaque formula early on with example after example that really just describes the inputs without ever saying how they are used. To make matters worse there is chapter after chapter of what is basically two bunches of statisticians arguing and Bayesian statistics sort of being used in rather uninspiring circumstances. It only really came alive for me when the author was describing its use in the hunt for mislaid nuclear weapons  and even then it is not at all clear how the technique was used from the way she describes it.
Most frustrating of all is that the second appendix contains a very clear example of a simple Bayesian working with a remarkable result. This is the first time in the whole book that it becomes fairly obvious what is going on with Bayesian statistics. This example should have been right up front, not in an appendix that half the readers won't even bother with, and there should have been similarly clear examples of some of the more complex applications. Not in full detail, but enough to get a feel for what is happening.
Overall, then, it seems the publishers who didn't want me to write about this made the correct call. I am the ideal audience  I worked in operational research, for goodness sake. And I still found most of it uninspiring and hard to understand how Bayesian methods were being used in the particular examples. What a shame.
Review by Brian Clegg
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Last update 16 April 2011