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Bayes' Rule - James V. Stone ***

Of all the areas of mathematics, probability is arguably the most intriguing to the non-mathematician, and this is particularly the case with Bayesian analysis, which can be delightfully counter-intuitive. However, the more complex aspects can be tricky to get your head around, so I was delighted to have the chance to read this book, subtitled 'a tutorial introduction to Bayesian analysis.'

 I need to say straight away that this isn't really a popular science title, and the author is very clear about this - it's a kind of textbook lite - but if you have found out a bit about Bayes this book is an opportunity to dive into it a little deeper without taking on the full rigour of a textbook approach. Why should you care? Bayes gives us a mechanism that enables us to do things like go from a known piece of information like 'what's the probability of a symptom given a disease' to estimate a much more interesting unknown like 'what's the probability of the disease given a symptom' - an extremely powerful mechanism.

 James Stone does his best to accommodate us ordinary folk. The book opens well, apart from a bizarrely heavy smattering of references on page 1, with a gentle introduction, and keeps the mood light after the classic disease application by looking for a mechanism of determining whether some said 'four candles' or 'fork handles' in the Two Ronnies style. If you are prepared to make an effort, for most of us probably a considerable effort, you will go on to pick up a lot more about using Bayes than you already knew (if you aren't a mathematician).

 It is rather unfortunate for the general reader, though, that the book obeys the rules of the textbook rather than a popular science exposition. This comes across in unnecessary use of terminology - defining things that, frankly we don't need to know - and in rapidly moving to using symbols in equations, where they are rarely necessary at this level and all they do is put readers off. I suspect the moment that Stone introduced the Greek letter theta (θ) he made things ten times harder - unless you do this kind of thing every day, suddenly the text gets far less readable - the eyes bounce off it.

 Even though I enjoyed the fork handles, I also thought the choice of examples could have been better. It was okay to use disease and symptom once, as it's an important application, but most of us rarely have to deal with this kind of situation and it would have been better to use more personally relevant applications. It was also unfortunate that when explaining random variables Stone chose a coin which is 90% likely to be heads and 10% likely to be tails - there is too much baggage attached to coins being 50:50. It would have been less confusing to have something that we might encounter (a scratch card, say) that is likely to be one value 90% of the time and the other 10%.

 If you make it to the final chapter you are rewarded with a very readable, if too brief, introduction to the distinction between Bayesian and frequentist approaches, and just a touch of the mind bending capabilities of Bayesian thinking. With a bit more of this contextual material throughout the experience would have been gentler and more enjoyable - but even as a closer to the book it provides interesting material.

 Don't expect, then that this book will make fun, popular science bedtime reading. It's not that kind of exercise. However, if you are prepared to overcome the onslaught of thetas and don't mind reading some statements several times to get what's being said, it is an excellent way to expand a vague understanding into a more sound knowledge of the basic mechanics of Bayesian analysis.
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See one of the mind-bending implications of Bayes' rule in our feature.
Review by Brian Clegg
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