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Zeros to heroes: Bayes's probability puzzle
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Bayes’s theorem AN ENGLISH cleric pondering balls on a billiard table is the improbable origin of one of the most powerful techniques in modern science. At its root is a simple question. Yet the answer, first outlined almost 250 years ago, provokes debate even now. In 1764, the Royal Society in London published a paper by Thomas Bayes, a Presbyterian minister and amateur mathematician, which addressed a tricky problem in the theory of probability. Till then, mathematicians had focused on the familiar problem of working out what to expect from, say, a tossed die, when one knows the chance of seeing a particular face is 1 in 6. Bayes was interested in the flip side: how to turn observations of an event into an estimate of the chances of the event occurring again. In his paper, Bayes illustrated the problem with an esoteric question about the location of billiard balls rolled onto a table. He came up with a formula that turned observations of their final locations into an estimate of the chances of future balls following them. All very trivial – except that the same basic issue underpins science: how do we turn observations into evidence for or against our beliefs? In other words, his work enabled observations to be used to infer the probability that a hypothesis may be true. Bayes was thus laying the foundations for the quantification of belief. But there was a problem; Bayes himself recognised it, and it still sparks controversy. To derive his formula, Bayes had to make assumptions about the behaviour of the balls, even before making the observations. He believed these so-called “priors” were reasonable, but could see that others might not. He wasn’t wrong. For most of the past 200 years, the application of Bayes’s methods to science have been dogged by arguments over this issue of prior assumptions. In recent years, scientists have become more comfortable with the idea of priors. As a result, Bayes’s methods are becoming central to scientific progress in fields ranging from cosmology to climate science (Nature, vol 350, p 371). Not bad for a formula describing the behaviour of billiard balls. Robert Matthews
11 September 2010 | NewScientist | 35
Hulton-Deutsch Collection/CORBIS
Bayes’s theorem AN ENGLISH cleric pondering balls on a billiard table is the improbable origin of one of the most powerful techniques in modern science. At its root is a simple question. Yet the answer, first outlined almost 250 years ago, provokes debate even now. In 1764, the Royal Society in London published a paper by Thomas Bayes, a Presbyterian minister and amateur mathematician, which addressed a tricky problem in the theory of probability. Till then, mathematicians had focused on the familiar problem of working out what to expect from, say, a tossed die, when one knows the chance of seeing a particular face is 1 in 6. Bayes was interested in the flip side: how to turn observations of an event into an estimate of the chances of the event occurring again. In his paper, Bayes illustrated the problem with an esoteric question about the location of billiard balls rolled onto a table. He came up with a formula that turned observations of their final locations into an estimate of the chances of future balls following them. All very trivial – except that the same basic issue underpins science: how do we turn observations into evidence for or against our beliefs? In other words, his work enabled observations to be used to infer the probability that a hypothesis may be true. Bayes was thus laying the foundations for the quantification of belief. But there was a problem; Bayes himself recognised it, and it still sparks controversy. To derive his formula, Bayes had to make assumptions about the behaviour of the balls, even before making the observations. He believed these so-called “priors” were reasonable, but could see that others might not. He wasn’t wrong. For most of the past 200 years, the application of Bayes’s methods to science have been dogged by arguments over this issue of prior assumptions. In recent years, scientists have become more comfortable with the idea of priors. As a result, Bayes’s methods are becoming central to scientific progress in fields ranging from cosmology to climate science (Nature, vol 350, p 371). Not bad for a formula describing the behaviour of billiard balls. Robert Matthews
11 September 2010 | NewScientist | 35