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Fisher and Mathematical Statistics Fisher and Mathematical Statistics John Aldrich University of Southampton R. A. Fisher Commemoration RSS June 2012 Fisher and Mathematical Statistics In 1923 Ronald Fisher proposed that Cambridge University create a Chair in Mathematical Statistics. Four decades later and largely through his influence mathematical statistics was an established discipline, particularly in the United States. But Fisher did not like the result “I believe sanity and realism can be restored to the teaching of Mathematical Statistics most easily and directly by entrusting such teaching largely to men and women who have had personal experience of research in the Natural Sciences.” (1956) I trace Fisher’s disenchantment by recalling his relations with eight individuals. Parts of these personal stories appear in Joan Fisher Box’s R. A. Fisher: The Life of a Scientist though mainly as background noise. Some relevant letters appear in J. H. Bennett’s Statistical Inference and Analysis. The bigger story involves changes in the nature of mathematics in the course of the twentieth century and in the scientific power of the United States. The mathematical statisticians— emphasising American connections Harold Hotelling Samuel Wilks Jerzy Neyman Daniel Dugué Harald Cramér Abraham Wald John Tukey Jimmie Savage. Fisher’s mathematics training In the nineteenth century Cambridge mathematics had been centred on applied mathematics—mathematical physics. Fisher recalled that as an undergraduate he was “out of sympathy with the recent passage of control in mathematical teaching from the earlier tradition of mathematical physicists to a school of pure mathematicians of largely continental derivation.” How Fisher placed statistics “The science of statistics is essentially a branch of Applied Mathematics and may be regarded as mathematics applied to observational data.” (1925) Through the 20s Fisher the applied mathematician worked alone on The theory of estimation (centred on maximum likelihood) The family of distributions related to the normal distribution The analysis of variance and design of experiments. Pure and applied (in 1922) I should gladly have withheld publication until a rigorously complete proof could have been formulated BUT the number and variety of the new results which the method discloses press for publication I am not insensible of the advantage which accrues to Applied Mathematics from the co-operation of the Pure Mathematician, and this co-operation is not infrequently called forth by the very imperfections of writers on Applied Mathematics. Yet … While Fisher admitted the possibility of perfecting the imperfect he did not recognise that in his case anybody had achieved this. In different ways Burnside, Irwin and Wishart thought his distribution theory arguments needed perfecting. Stigler’s “epic story” of maximum likelihood was a straightening out of the arguments in Fisher 1922. But Fisher wasn’t interested in the sequel. My first group: pure mathematicians Fisher was comfortable with Harold Hotelling Daniel Dugué Harald Cramér Harold Hotelling (1895-1973) Princeton PhD in topology Reviewed Statistical Methods for Research Workers for JASA on his own initiative: “The author's work is of revolutionary importance and should be far better known in this country.” Went to Rothamsted as a “voluntary worker” in 1929 A possible collaborator Collaboration projected on a mathematical version of Statistical Methods with Hotelling providing the proofs. Hotelling and Fisher shared results on maximum likelihood. Collaboration fizzled out in the late 30s The much younger Daniel Dugué (1912- 1987) Dugué, a student of Darmois, sent Fisher his thesis, Application des propriétés de la limite au sens du calcul des probabilités a l'étude de diverse questions d'estimation (1937) The thesis married Khinchin and Kolmogorov on limit theorems to Fisher on maximum likelihood. Fisher and Dugué Fisher: “I am indeed glad that this interesting subject is now receiving such acute and careful analysis as that of your papers.” Dugué came to London as a Rockefeller fellow in 1937/8. Fisher on Dugué, “an excellent mathematician, a polite and very well-bred boy, who has apparently never seen a computing machine before and hesitates a little to dirty his hands with one.” Dugué learns to use a calculating machine and acquires a new maître. He stops working his earlier vein—cured perhaps. Fisher learns more about Continental probability. But not enough to improve his opinion of it. Harald Cramér (1893-1985) number theorist, actuary and probabilist In 1937 published Random Variables and Probability Distributions dubbed “the first modern book on probability in English” He met Fisher in 1939 recalling their conversation: I had expressed my admiration for his geometrical intuition in dealing with probability distributions in multidimensional spaces, and received the somewhat acid reply: “I am sometimes accused of intuition as a crime!” In 1946 Princeton published his Mathematical Methods of Statistics Mathematical Methods Cramér: Fisher (more in sorrow than in anger?): I recently received a very highbrow treatise of a first class Swede, namely Cramer, purporting to deal with mathematical statistics, of which I think a full half was a comprehensive introduction to a theory of point sets. I suppose in view of the didactic manner in which the theory of probability is approached in Russia, for example, and in France, and in recent years in the United States, this sort of thing must seem a quite essential clarification of the subject. The second group: “academic mathematicians from abroad” Samuel Wilks (1906 - 1964) Jerzy Neyman (1894-1981) These aspired to correct Fisher. In 1932/3 Wilks came to learn from E. S. Pearson and Wishart Fisher read his paper on the analysis of variance First reaction I am much puzzled as to why you should feel that such an elementary point as that you discuss should need a new and very elaborate discussion. Final view I judge that the whole work was undertaken under a most regrettable misapprehension, and that you may need some little time to familiarise yourself with methods of reasoning and demonstration other than that in which you have so far specialised. Neyman became part of the London scene in 1934/5 Initially Fisher took a benign interest: it had been of “great interest to me to follow the attempts which Drs Neyman and Pearson have made to develop a theory of estimation independent of some of the concepts I have used.” His attitude changed when Neyman began criticising Fisher’s theory of experimental design. 1935 Design of Experiments sets a new tone Experimenters pitted against mathematicians The experimenter assumes that it is possible to draw valid inferences from the results of experimentation… Many mathematicians ... would say that it is not possible... The notion that different tests of significance are appropriate to test different features of the same null hypothesis presents no difficulty to workers engaged in practical experimentation, but has been the occasion of much theoretical discussion among statisticians. Location shifts to America Fisher did not visit the US between 1936 and 1946. In that decade mathematical statistics took off. Hotelling, Wilks and Neyman (who moved to Berkeley in 1938) were all important in this development. As in Britain, there were many new entrants as the War diverted mathematicians into statistics. Wilks and the Annals of Mathematical Statistics The Annals was founded in 1930 but in its early years published little of any note. Wilks moved to Princeton in 1933 From 1939 Wilks edited the journal and “transformed it into the most influential statistics journal in the world.” (Stigler) The Annals of 1939 Modelled on E. S. Pearson’s Biometrika International editorial board. Contributions from Bartlett and Welch as well as from Fisher. The exigencies of war Leading Annals figures of the 40s William Feller, Walter Shewhart, Samuel Wilks, Paul Dwyer, Abraham Wald and Harold Hotelling Abraham Wald (1902-1950) A geometer who worked on economic statistics in Vienna In 1938 he emigrated to the United States where statistical mentor was Harold Hotelling. In 1939 published his first work on statistical decision theory. Fisher owned a copy of Statistical Decision Functions and did not like it. There appears to have been no contact between the two. Joan Box on Fisher & American statisticians When he returned to the United States in 1946, he was welcomed by the younger statisticians as a great originator and authority certainly but also as a foreigner whose ways were not always their ways, nor his thoughts their thoughts. The transition had begun from the reality to the myth from acceptance of Fisher as an exciting colleague in their research to his reception as an oracle, of uncertain temper and controversial meaning. These points are illustrated in Fisher’s relations with John W. Tukey (1915-2000) a Princeton topologist like Hotelling—though topology had changed. His mentor in statistics was Wilks. Leonard Jimmie Savage (1917-71) a geometer who came under the influence of von Neumann and Wald. His statistical mentors were Milton Friedman and W. Allen Wallis, students of Hotelling. All admired Fisher. They did not want Fisher’s views on their projects and generally Fisher was not interested in those projects. They wanted to get to the bottom of his views on Fiducial inference The problem of the Nile The role of alternative hypotheses in testing. Reviewing the 11th edition of Statistical Methods Tukey said that most statisticians should begin their education elsewhere but “they should feel that until they understand why Fisher says and computes what he does, in this book and in The Design of Experiments, they are not equipped with one of their basic tools.” Fisher’s exasperation as he and Tukey correspond on fiducial inference 1954-8 If you must write about someone else’s work, it is, I feel sure, worth taking even more than a little trouble to avoid misrepresenting him. … as you would see if ever you got your bull- headed mind to stop and think.
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