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 sequentiai Bayesian procedure (anticipating sequential analysis) which was used from the sorine of 1941 until the middle of 1943. It was invented mainlv bv A. M. Turina and was perhaps the first important sequential Bayesian IE is unnecessab to describe it here. Before Banburismus could be started on a given day it was necessary to identifv which of nine ‘biaram’ (or ‘diaraph’) tables was in use on that day. In Turing’s approach to this identification hk had io istimate the probabilities of certain ‘trigraphs’. rrhese trigraphs were used. as described below. for determinine the initial wheel settings of messages). For estimatidg the probabilities, Turing inventedin important special case o the nonparametric (nonhypermetric) Empirid Bayes method independently of Herbert Robbins. The techniaue is the sumxisine form of Emdrical Baves in which a physical prior is assumed to eist but no apbroxiGate functional fonn is assumed for it.     

Choosing between two unknown intervals: an empirical bayes testing approach to a classification problem

Let X₁,X₂…be i.i.d. observations from a mixture density. The support of the unknown prior distribution is the union of two unknown intervals. The paper deals with an empirical Bayes testing approach (?≤ c against>c where c is an unknown parameter to be estimated) in order to classify the observed variables as coming from one population or the other as ? belongs to one or the other unknown interval. Two methods are proposed in which asymptotically optimal decision rules are constructed avoiding the estimation of the unknown prior. The first method deals with the case of exponential families and is a generalization of the method of Johns and Van Ryzin (1971, 1972) whereas the second one deals with families that are closed under convolution and is a Fourier method. The application of the Fourier method to some densities (i.e. contaminated Gaussian distributions, exponential distribution, double-exponential distribution) which are interesting in view of applications and which cannot be studied by means of the direct method, is also considered herein.     

Regression analysis of hierarchical poisson-like event rate data: superpopulation model effect on predictions

This paper studies prediction of future failure (rates) by hierarchical empirical Bayes (EB) Poisson regression methodologies. Both a gamma distributed superpopulation as well as a more robust (long-tailed) log student-t superpopulation are considered. Simulation results are reported concerning predicted Poisson rates. The results tentatively suggest that a hierarchical model with gamma superpopulation can effectively adapt to data coming from a log-Student-t superpopulation particularly if the additional computation involved with estimation for the log-Student-t hierarchical model is burdensome.     

Simultaneous estimation of gamma means using a hierarchical generalized linear model

The problem of simultaneously estimating p Gamma means is investigated when the means are believed a priori to satisfy an r-dimensional generalized linear model. Using a Bayesian hierarchical model to reflect the uncertainty in the linear model, approximate methods are proposed to compute the posterior densities. The resulting estimator shrinks the usual estimator toward a prior estimator where the size of the shrinkage depends upon the agreement of the observed data with the proposed generalized linear model.