Turing’s Bayesian contributions

Following The Imitation Game, this recent movie about Alan Turing played by Benedict “Sherlock” Cumberbatch, been aired in French theatres, one of my colleagues in Dauphine asked me about the Bayesian contributions of Turing. I first tried to check in Sharon McGrayne‘s book, but realised it had vanished from my bookshelves, presumably lent to someone a while ago. (Please return it at your earliest convenience!) So I told him about the Bayesian principle of updating priors with data and prior probabilities with likelihood evidence in code detecting algorithms and ultimately machines at Bletchley Park… I could not got much farther than that and hence went checking on Internet for more fodder.

“Turing was one of the independent inventors of sequential analysis for which he naturally made use of the logarithm of the Bayes factor.” (p.393)

I came upon a few interesting entries but the most amazìng one was a 1979 note by I.J. Good (assistant of Turing during the War) published in Biometrika retracing the contributions of Alan Mathison Turing during the War. From those few pages, it emerges that Turing’s statistical ideas revolved around the Bayes factor that Turing used “without the qualification `Bayes’.” (p.393) He also introduced the notion of ban as a unit for the weight of evidence, in connection with the town of Banbury (UK) where specially formatted sheets of papers were printed “for carrying out an important classified process called Banburismus” (p.394). Which shows that even in 1979, Good did not dare to get into the details of Turing’s work during the War… And explains why he was testing simple statistical hypothesis against simple statistical hypothesis. Good also credits Turing for the expected weight of evidence, which is another name for the Kullback-Leibler divergence and for Shannon’s information, whom Turing would visit in the U.S. after the War. In the final sections of the note, Turing is also associated with Gini’s index, the estimation of the number of species (processed by Good from Turing’s suggestion in a 1953 Biometrika paper, that is, prior to Turing’s suicide. In fact, Good states in this paper that “a very large part of the credit for the present paper should be given to [Turing]”, p.237), and empirical Bayes.