Turing's anticipation of empirical bayes in connection with the cryptanalysis of the naval enigma

The Enigma was a cryptographic (enciphering) machine used by the German military during WWII. The German navy changed part of the Enigma keys every other day. One of the important cryptanalytic attacks against the naval usage was called Banburismus, a sequential Bayesian procedure (anticipating sequential analysis) which was used from the spring of 1941 until the middle of 1943. It was invented mainly by A. M. Turing and was perhaps the first important sequential Bayesian procedure. It is unnecessary to describe it here. Before Banburismus could be started on a given day it was necessary to identify which of nine 'bigram' (or 'digraph') tables was in use on that day. In Turing's approach to this identification he had to estimate the probabilities of certain 'trigraphs'. (These trigraphs were used, as described below, for determining the initial wheel settings of messages.) For estimating the probabilities, Turing invented an important special case of the nonparametric (nonhyperparametric) Empirical Bayes method independently of Herbert Robbins. The technique is the surprising form of Empirical Bayes in which a physical prior is assumed to exist but no approximate functional form is assumed for it.