I. J. Good

The Maximum Entropy Formalism.

An exchangeable or removable nozzle arrangement for use in a fluidized bed furnace is movable and in sealing contact with a surrounding sleeve at one end of the nozzle and may be withdrawn from the sleeve through a valve at the other end of the sleeve. An inlet to the space between the sleeve and the nozzle is connected to a source of pressurized fluidizing gas. Upon removal of a nozzle, while the furnace is under load, the bed is maintained in its fluidized state.

Speculations Concerning the First Ultraintelligent Machine

Publisher Summary An ultra-intelligent machine is a machine that can far surpass all the intellectual activities of any man however clever. The design of machines is one of these intellectual activities; therefore, an ultra-intelligent machine could design even better machines. To design an ultra-intelligent machine one needs to understand more about the human brain or human thought or both. The physical representation of both meaning and recall, in the human brain, can be to some extent understood in terms of a subassembly theory, this being a modification of Hebbs cell assembly theory. The subassembly theory sheds light on the physical embodiment of memory and meaning, and there can be little doubt that both needs embodiment in an ultra-intelligent machine. The subassembly theory leads to reasonable and interesting explanations of a variety of psychological effects.

Density Estimation and Bump-Hunting by the Penalized Likelihood Method Exemplified by Scattering and Meteorite Data

Abstract The (maximum) penalized-likelihood method of probability density estimation and bump-hunting is improved and exemplified by applications to scattering and chondrite data. We show how the hyperparameter in the method can be satisfactorily estimated by using statistics of goodness of fit. A Fourier expansion is found to be usually more expeditious than a Hermite expansion but a compromise is useful. The best fit to the scattering data has 13 bumps, all of which are evaluated by the Bayesian interpretation of the method. Eight bumps are well supported. The result for the chondrite data suggests that it is trimodal and confirms that there are (at least) three kinds of chondrite.

The fractional dimensional theory of continued fractions

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.

Some history of the hierarchical Bayesian methodology

SummaryA standard technique in subjective “Bayesian” methodology is for a subject (“you”) to make judgements of the probabilities that a physical probability lies in various intervals. In the hierarchical Bayesian technique you make probability judgements (of a higher type, order, level, or stage) concerning the judgements of lower type. The paper will outlinesome of the history of this hierarchical technique with emphasis on the contributions by I. J. Good because I have read every word written by him.

The Bayes/non-Bayes compromise : a brief review

Abstract Various compromises that have occurred between Bayesian and non-Bayesian methods are reviewed. (A citation is provided that discusses the inevitability of compromises within the Bayesian approach.) One example deals with the masses of elementary particles, but no knowledge of physics will be assumed.

Significance Tests in Parallel and in Series

Abstract The advice is often given that significance tests should be selected before sampling evidence is examined. It is suggested here that this advice is appropriate only for inexperienced statisticians, and an approximate rule of thumb is tentatively proposed in the hope of provoking discussion, namely that the statistician could in some cases use a harmonic mean or weighted harmonic mean of the tail-area probabilities arising from various tests, all on the same evidence (tests in “parallel”). This rule of thumb should not be used if the statistician can think of anything better to do, and especially of course if he is in a position to use a sufficient statistic (or an “efficacious” one, in a sense defined below). An application is given to the judgment of the weights that may be used for combining tests in series.

Exact Distributions for χ2 and for the Likelihood-Ratio Statistic for the Equiprobable Multinomial Distribution

Abstract A sample of size N is taken from an equiprobable t-category multinomial distribution. The precise values of Pearsons X 2 and the likelihood ratio statistic A were computed for t = 3(1)6, N = 3(1)12; t = 6(1)14, N = 6(1)2t; t=15(1)18, N = 6(1)28. The logarithms of the tail areas were smoothed and fitted by means of quadratic expressions. Much information is given concerning the accuracy of the chi-squared fits to the distributions of X 2 and A. A table of −log10 P(χ2 > a) is given, for a = 1(1)50, and d.f. = 1(1)20. The distribution of the total number of repeats within cells is also discussed.

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.

The Bayes/Non-Bayes Compromise and the Multinomial Distribution

Abstract Compromises between Bayesian and non-Bayesian significance testing are exemplified by examining distributions of criteria for multinominal equiprobability. They include Pearsons X2, the likelihood-ratio, the Bayes factor F, and a statistic G that previously arose from a Bayesian model by “Type II Maximum Likelihood.” Its asymptotic distribution, implied by the theory of the “Type II Likelihood Ratio,” is remarkably accurate into the extreme tail. F too can be treated as a non-Bayesian criterion and is almost equivalent to G. The relationship between F and its own tail area sheds further light on the relationship between Bayesian and “Fisherian” significance.