The Art of Statistics: Learning from Data

by David Spiegelhalter

So let’s put all this together in Bayes’ theorem, which simply says that the initial odds for a hypothesis × the likelihood ratio = the final odds for the hypothesis For the doping example, the initial odds for the hypothesis ‘the athlete is doping’ is 1/49, and the likelihood ratio is 19, so Bayes’ theorem says the final odds are given by 1/49 × 19 = 19/49 These odds of 19/49 can be transformed to a probability of 19/(19+49) = 19/68 = 28%. So this probability, which was obtained from the expected frequency tree in a rather simple way, can also be derived from the general equation for Bayes’ theorem. In more technical language, the initial odds are known as the ‘prior’ odds, and the final odds are the ‘posterior’ odds. This formula can be repeatedly applied, with the posterior odds becoming the prior odds when introducing new, independent items of evidence. When combining all the evidence, this process is equivalent to multiplying the independent likelihood ratios together to form a composite likelihood ratio. Bayes’ theorem looks deceptively basic, but turns out to encapsulate an immensely powerful way of learning from data.

Bayes odds