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Words in his mouth

In the course of reviewing Doug Hubbard’s book on failing risk management, I thought it was worth a little wanderette on how much Thomas Bayes has been abused, or bigged up depending on how you see it.  I thought it would go as follows.

“Bayes’ contribution to posterity was to get his name against a formula for joint probabilities which verges on the bleeding obvious: p(A,B)=p(A|B)p(B)=p(B|A)p(A).  Maybe it was a bit less obvious when people were struggling to understand randomness in the eighteenth century.

“The great triumph was to use the formula when one of the sample space events was changed into a hypothesis.  With a degree of belief interpretation of the probability of a hypothesis you can use the formula to get p(H|D)=p(D|H)p(H)/p(D) and find the probability conditional on the observed data.  Bayesian inference was born, along with the idea that you can update your beliefs as more data emerges.  While Bayesian inference is an elegant technique (though I sometimes wonder what the other bits add to the likelihood function L(H|D)=p(D|H) – more on this another time) it has nothing to do with Bayes; it took 200 years to make this conceptual leap.

“Finally once the idea of Bayesian inference has made subjective probabilities respectable, it’s a short step to call them Bayesian probabilities, a shocking and erroneous extrapolation the not-so-great man would never have approved of.”

Turns out I was wrong.  His posthumous paper was exactly concerned with discovering the probability of a binomial parameter based on data, in his case the results of a lottery.  He was able to provide a frequentist interpretation, but the intention is pure Bayesian inference after all.  It did indeed take 200 years to start to get it rationalised though and you still shouldn’t call them Bayesian probabilities.

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