Harold Jeffreys as a Statistician

 

 

The Jeffreys prior is part of the furniture of Bayesian statistics but otherwise the work of Harold Jeffreys is little known. Jeffreys was a noted physical scientist who re-established the statistical theory of his time on Bayesian foundations. This page is a guide to literature and websites which may be useful to anyone interested in Jeffreys’s statistical work and its background. The emphasis is on Jeffreys’s own writings and on the older literature.

 

John Aldrich, University of Southampton, Southampton, UK. (home)

 

Original version February 2003

Most recent changes November 2011


Introduction

   Scientific inference

  Controversy with Fisher

   Theory of Probability

   After Probability

 

 

 

 

Links

This guide has links to some free sites, including MacTutor for biographies and the University of Adelaide Library for R. A. Fisher material. There are also links to the institutional subscription service JSTOR: for information go to http://www.jstor.org/.

 

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Introduction

Around 1930 Harold Jeffreys F.R.S. (1891-1989), an established writer on physics, applied mathematics and scientific method, began applying his version of the Bayesian argument to statistics. Over the next decade or two he produced more than 20 articles on statistics and a large book, Theory of Probability.

 

Jeffreys gave a succinct account of what he did in probability/statistics and why he did it in

  • H. Jeffreys (1974) Fisher and Inverse Probability, International Statistical Review, 42, (1), 1-3.

Inverse probability was how Jeffreys referred to the Bayesian approach; the terms Bayesian, classical, and frequentist only became current after his main work was done.  There are many relevant entries on the Earliest known uses pages, see Probability & Statistics on the Earliest Uses Pages: a search for Jeffreys on the front page will show on which of the individual pages he appears and a further search on the page will bring up all his appearances.

 

Earlier writers used Bayesian methods but Jeffreys has a good claim to be considered the first Bayesian statistician in that he used only Bayesian methods. Earlier Bayesian writing—from authors such as Laplace, Gauss, Edgeworth and Pearson—is described in

  • Andrew I. Dale (1999) A History of Inverse Probability from Thomas Bayes to Karl Pearson, second edition, New York: Springer-Verlag.
  • Anders Hald (1998) A History of Mathematical Statistics 1750-1930, New York: Wiley.
  • J. Pfanzagl & O. Sheynin (1996) Studies in the History of Probability and Statistics XLIV A Forerunner of the t-Distribution, Biometrika, 83, 891-898. (available through JSTOR)

Jeffreys’s knowledge of this early literature was limited and he spent some time re-inventing the wheel.

 

Life

There is a short MacTutor biography focussing on Jeffreys’s activities as an applied mathematician—it has photographs and conveys something of his personality. There are encyclopedia articles on Jeffreys as a statistician by Lindley and Zellner. Cook’s (1991) memoir concentrates on Jeffreys’s contributions to physics and applied mathematics. The festschrift edited by Zellner includes accounts of Jeffreys’s contribution to statistics by Good, Geisser and Lindley. There are memorial pieces in Chance including one by Lady Jeffreys (Bertha Swirles Jeffreys, Bertha Swirles, Lady Jeffreys Bertha Jeffreys), his wife and collaborator on Methods of Mathematical Physics. (available on Internet Archive.  Howie’s book on the controversy with Fisher discusses other phases of Jeffreys’s life and is particularly strong on his work in the 1920s. Aldrich looks at Jeffreys’s preparation for writing the Theory of Probability.

  • Dennis V. Lindley (1998) Sir Harold Jeffreys Encyclopedia of Biostatistics 3, 2124-2125. Chichester: Wiley.
  • A. Zellner (2001) Jeffreys, Sir Harold (1891-1989) International Encyclopedia of the Social and Behavioral Sciences, 7960-7963. Kidlington, Oxford: Pergamon.
  • A. H. Cook (1991) Sir Harold Jeffreys, Biographical Memoirs of Fellows of the Royal Society, 37, 303-331.
  • Arnold Zellner, (ed.) (1980) Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys, Amsterdam: North-Holland.
  • Seymour Geisser (1980) The Contributions of Sir Harold Jeffreys to Bayesian Inference, pp. 13-20 of Zellner (1980).
  • Irving John Good (1980) The Contributions of Jeffreys to Bayesian Statistics, pp. 21-34 of Zellner (1980).
  • Dennis V. Lindley (1980) Jeffreys’s Contribution to Modern Statistical Thought, pp. 35-40 of Zellner (1980).
  • B. Swirles (Lady Jeffreys) (1991) Harold Jeffreys: Some Reminiscences, Chance, 4 (2), 22-26.
  • D. V. Lindley (1989) Obituary: Harold Jeffreys, 1891-1989, Journal of the Royal Statistical Society A, 152, 417-418. JSTOR
  • D. V. Lindley (1991) Sir Harold Jeffreys, Chance, 4 (2), 10-14 & 21.
  • D. Howie (2002) Interpreting Probability: Controversies and Developments in the Early Twentieth Century, New York, Cambridge University Press. (Contents and introductory chapter are available. Reviews by Howson and Fitelson and Aldrich are also available on line.)
  • J. Aldrich (2005) The Statistical Education of Harold Jeffreys, International Statistical Review, 73(2), 289-308.

 

Photographs. The MacTutor biographies of Jeffreys and his wife have photographs. The four photos of Jeffreys at the Emilio Segrè Visual Archives include one at the time of their wedding.

 

Video. The Royal Statistical Society created a series of filmed interviews with distinguished statisticians: Jeffreys was interviewed by Lindley. Copies can be purchased from the Society.

 

The Jeffreys papers at St. John’s College have been catalogued and the catalogue is available here.

 

Jeffreys wrote more than 400 papers, usually alone, on subjects ranging across celestial mechanics, fluid dynamics, meteorology, geophysics and probability: see here for a list. There is a six volume set of papers:

·        H. Jeffreys and B. Swirles (eds.) (1971-77) Collected Papers of Sir Harold Jeffreys on Geophysics and other Sciences in six volumes, London, Gordon & Breach.

The statistics papers are in volume 6, Mathematics, Probability & Miscellaneous Other Sciences. The coverage is not comprehensive for Jeffreys omitted papers that had been superseded by his books Scientific Inference and Theory of Probability.

 

Jeffreys is often described as the founder of modern British geophysics. Many of his contributions are summarised in his book The Earth. Brush (1980) describes one of Jeffreys’s main discoveries, that the core of the Earth is liquid. Jeffreys figures prominently in Brush’s three volume A History of Modern Planetary Physics (Cambridge University Press 1996).

·        H. Jeffreys (1924) The Earth, further editions in ‘29, ’52, ’59, ’70 and ’76, Cambridge, Cambridge University Press.

  • Stephen G. Brush (1980) Discovery of the Earth’s Core, American Journal of Physics, 48, 705-724.

·        E. R. Lapwood (1982) Contributions of Sir Harold Jeffreys to Theoretical Geophysics, Mathematical Scientist, 7, 69-84.

·        B. A. Bolt (1991) Sir Harold Jeffreys and Geophysical Inverse Problems, Chance, 4, 15-17.

Jeffreys talked about his work in seismology in an interview with Henry Spall. Jeffreys was elected to the Royal Society in 1925 and the election certificate shows how he was regarded at the time.

 

Jeffreys spent virtually his entire life in Cambridge, as a student and then successively a College lecturer in mathematics in 1922, Reader in Geophysics in 1931 and Plumian Professor of Astronomy and Experimental Philosophy in 1946. Cambridge mathematics provided many of the leading figures in British statistics and Jeffreys’s position gave him access to undergraduates of the calibre of Finney, Lindley, Cox, Durbin, …. However his influence on their young minds seems to have been limited. The quality of his lecturing was perhaps a factor—see MacTutor or Tuzo Wilson—although that did not prevent him attaining a commanding position in geophysics. His only PhD student in statistics was V. S. Huzurbazar.

 

A.W.F. Edwards tells me that Jeffreys’s influence on Barnard was considerable and that Jeffreys probably influenced Alan Turing. I. J. Good does not discuss the question of influence in hisA. M. Turing’s Statistical Work in World War II” (Biometrika, 66, (1979), pp. 393-396 JSTOR) but there seem to be clear echoes of Jeffreys in Turing’s thinking.

 

 

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Scientific Inference  back to contents

Jeffreys first used probability to deal with problems in the philosophy of science. Jeffreys read Karl Pearson’s Grammar of Science in 1914 and it made a great impression on him, with its emphasis on the probabilistic basic of scientific inference. Jeffreys treated probability as a degree of reasonable belief, an epistemic conception common to several Cambridge philosopher/logicians, including W. E. Johnson, J. M. Keynes and C. D. Broad. That conception reached Jeffreys, through his collaborator Dorothy Wrinch (Wikipedia MacTutor) who had attended Johnson's lectures. Wrinch and Jeffreys used probability to explicate induction and investigate the reasonableness of scientific theories, including general relativity. For the purpose of appraising scientific theories John Venn’s probability as a limiting frequency was useless but Wrinch and Jeffreys considered it mathematically unsound as well; Venn’s Logic of Chance is available from GDZ.

  • D. Wrinch & H. Jeffreys (1919) On Some Aspects of the Theory of Probability, Philosophical Magazine, 38, 715-731.         
  • D. Wrinch & H. Jeffreys (1921/23) On Certain Fundamental Principles of Scientific Inquiry (Two Papers), Philosophical Magazine, 42, 369-390, 45, 368-374.

Howie is very informative on the Wrinch-Jeffreys collaboration. Aldrich (2005)  also discusses this phase of Jeffreys’s career.

 

Jeffreys summarised and extended the Wrinch-Jeffreys work in his book Scientific Inference. 

  • H. Jeffreys (1931) Scientific Inference, reprinted with additions in ‘37 and with new editions in ‘57 and ‘73, Cambridge: Cambridge University Press. Full text of 1st edition on Internet Archive. Preface from MacTutor. Text of 3rd edition on Google books.

 

The first edition was noticed in the philosophy journals.  See

  • R. B. Braithwaite Mind (New Series), 40, (Oct., 1931), 492-501. (available through JSTOR)
  • Ernest Nagel Journal of Philosophy, 15, (Jul., 1932), 409-412. (available through JSTOR)
  • E. T. Mitchell Philosophical Review, 43, (Jan., 1934), 92-94. (available through JSTOR)

 

The second edition (which was essentially a new book) was noticed more widely.  See

·        Anon. Biometrika, 44, No. 3/4 (Dec., 1957), 538-539. (available through JSTOR)

  • Wesley C. Salmon Philosophy of Science, 24, (Oct., 1957), 364-366. (available through JSTOR)
  • L. E. Palmieri Philosophy and Phenomenological Research, 18, (Dec., 1957). (available through JSTOR)
  • Stephen F. Barker Philosophical Review, 67, (Jul., 1958), 404-407. (available through JSTOR)
  • Arnold Koslow Journal of Philosophy, 57, (Jun. 9, 1960), 384-391. (available through JSTOR)
  • Michael Scriven Science, New Series, 126, (Aug. 16, 1957), 313. (available through JSTOR)
  • Patrick Suppes Journal of the American Statistical Association, 53, (Sep., 1958), 755-756. (available through JSTOR)
  • D. V. Lindley Journal of the Royal Statistical Society. Series A (General), 120, 2 (1957), 220-221. (available through JSTOR)

The third edition, more or less a second edition of the second edition, got some attention. See

  • V. Barnett Journal of the Royal Statistical Society. Series A (General), 137, (1974), 619-620. (available through JSTOR)
  • M. H. D. Journal of the American Statistical Association, 69, (Dec., 1974), 1052-1053 (available through JSTOR)

In the preface to the 2nd edition of the Probability Jeffreys explained his relationship to the other Cambridge writers. He appears not to have discussed probability with them. There is no record of any interaction with Broad or Johnson. They belonged to different colleges and were “moral scientists.”

According to Howie, Jeffreys met Keynes once—on a train. Jeffreys reviewed Keynes’s Treatise on Probability, welcoming it but faulting it for its caution.

·        The Theory of Probability Nature, 109, (1922), 132-133.

There is a brief account of Jeffreys’s view of Keynes in

·        J. Aldrich ((2008) Keynes among the Statisticians. History of Political Economy, 40, 265-316. Revised version of Southampton University Economics Discussion Paper 2006

 

Jeffreys knew Frank Ramsey (1903-1930) (N.-E. Sahlin St. Andrews D. H. Mellor) but did not know until after he had died death that he was interested in probability. Jeffreys discusses Ramsey’s views in the Theory of Probability.

 

Jeffreys did not know the work of the Italian subjectivist, Bruno de Finetti (1906-1985), although de Finetti wrote about Jeffreys’s Scientific Inference in a 1938 article translated as

  • Bruno de Finetti (1985) Cambridge Probability Theorists, Manchester School, 53, 348-363.

 

A recent article compares Jeffreys with Ramsey and de Finetti

  • Maria Carla Galavotti (2003) Harold Jeffreys’ Probabilistic Epistemology: between Logicism and Subjectivism, British Journal for the Philosophy of Science, 54, 43-57.

Placing Jeffreys has also been an issue in the statistical literature: see Zellner and Kass.

 

Philosophers of probability have generally taken Keynes as the representative Cambridge “objective Bayesian”. But Jeffreys appears in

  • Colin Howson (1995) Theories of Probability, British Journal for the Philosophy of Science, 46,  1-32. (available through JSTOR)

 

Bayesianism is now fashionable in philosophy of science. An early exponent building on Jeffreys was

  • Roger D. Rosencrantz (1977) Inference, Method and Decision: Towards a Bayesian Philosophy of Science, Dordrecht: Reidel.

 

These references relate to philosophy but Scientific Inference took the Wrinch & Jeffreys ideas on probability off in a new direction by considering how physical scientists used probability when they analysed data. Chapter V on Errors includes a Bayesian derivation of Student’s distribution.

 

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The controversy with Fisher   back to contents

Around 1930 Jeffreys, now concentrating on empirical geophysics, began devising methods for analysing data based on epistemic probability. He was extending the methods used by physical scientists and did not know much about, or greatly esteem, the efforts of statisticians. Meanwhile Ronald Fisher (1890-1962), probably the most influential twentieth century statistician, had rejected the Bayesian approach and based his work, including maximum likelihood, on frequentist foundations. There is a fine biography of Fisher by his daughter

  • Joan Fisher Box (1978) R. A. Fisher: The Life of a Scientist, New York: Wiley.

For more on Fisher see A Guide to R. A. Fisher.

 

Fisher and Jeffreys first took serious notice of each another in 1933. About all they knew of each other's work was that it was founded on a flawed notion of probability. Jeffreys (1933) criticised Fisher (1932) and Fisher (1933) criticised Jeffreys (1932) with a rejoinder, Jeffreys (1933a). The journal called a halt to the controversy by getting the parties to coordinate their last words, Fisher (1934) and Jeffreys (1934). (The Fisher articles are all available from Adelaide. Some pre-1938 volumes of the RS Proceedings are available from Gallica, but not 138.) Aldrich (2008) considers Fisher’s attitude to Bayes and to Bayesian inference.

·        R. A. Fisher (1932) Inverse Probability and the Use of Likelihood, Proceedings of the Cambridge Philosophical Society, 28, 257-261.

·        H. Jeffreys (1933) On the Prior Probability in the Theory of Sampling, Proceedings of the Cambridge Philosophical Society, 29, 83-87.

·        H. Jeffreys (1932) On the Theory of Errors and Least Squares, Proceedings of the Royal Society, A, 138, 48-55. (available through JSTOR)

·        R. A. Fisher (1933) The Concepts of Inverse Probability and Fiducial Probability Referring to Unknown Parameters, Proceedings of the Royal Society, A, 139, 343-348. (available through JSTOR)

·        H. Jeffreys (1933a) Probability, Statistics, and the Theory of Errors, Proceedings of the Royal Society, A, 140, 523-535 (available through JSTOR)

·         R. A. Fisher (1934) Probability, Likelihood and the Quantity of Information in the Logic of Uncertain Inference, Proceedings of the Royal Society, A, 146, 1-8. (available through JSTOR)

·        H. Jeffreys (1934) Probability and Scientific Method, Proceedings of the Royal Society, A, 146, 9-16. (available through JSTOR)

·        J. Aldrich (2008) R. A. Fisher on Bayes and Bayes’ Theorem. Bayesian Analysis, 3, 161-170.

     

 

Howie gives a full account of the dispute but Lane’s brief account is still useful. There is an even briefer account in Aldrich (2004). Seidenfeld considers how Jeffreys and Fisher used Keynes’s work in the debate.

  • D. A. Lane (1980), Fisher, Jeffreys and the Nature of Probability, pp. 148-160 in Fienberg, S. E. & D. V. Hinkley (1980) (eds.) R. A. Fisher: An Appreciation, New York, Springer.
  • J. Aldrich (2004) Harold Jeffreys and R. A. Fisher, ISBA Bulletin, 11, (June), 7-9.
  • T. Seidenfeld (1996) Jeffreys, Fisher and Keynes: Predicting the Third Observation Given the First Two, In New Perspectives on Keynes, edited by A. F. Cottrell & M. S. Lawlor: History of Political Economy, 27, (Supplement) 39-52.

Aldrich (2005) discusses the controversy but goes beyond it to consider how Jeffreys treated Fisher’s ideas in the Theory of Probability. Fisher did not mention modern work on inverse probability in his Statistical Methods for Research Workers (see Likelihood and Probability in Fisher’s Statistical Methods ) although Jeffreys asked him to—see correspondence p. 170.

 

Fisher and Jeffreys never accepted the validity of each other’s approach but their relationship mellowed into one of relaxed toleration. Their developing relationship can be followed in their letters which have been printed with notes in Bennett. There are more personal memories in Swirles and Box.

·        J. H. Bennett (1990) (ed.) Statistical Inference and Analysis: Selected Correspondence of R. A. Fisher, Oxford, University Press.

·        G. A. Barnard (1992) Review of Statistical Inference and Analysis: Selected Correspondence of R. A. Fisher (edited by J. H. Bennett). Statistical Science, 7, 5-12. (available through JSTOR)

 

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Theory of Probability   back to contents

In the years following Scientific Inference Jeffreys worked hard on statistics. In 1937 an enlarged edition of the book came out but he soon judged this to be inadequate, telling Fisher in September 1938, “I wish some public benefactor would subsidize [the publisher] to scrap Scientific Inference and give me a chance of writing something up to date.” Bennett (p. 170)

 

In 1939 he had a new book with a different publisher that incorporated the papers he had been writing on statistics. The Theory of Probability by Harold Jeffreys, Reader in Geophysics, University of Cambridge, appeared in the International Series of Monographs on Physics. The book is still in print as one of the Oxford Classic Texts in the Physical Sciences, although it seems always to have been read more by statisticians than by physicists.

 

  • Harold Jeffreys (1939) Theory of Probability, with a second edition in 1948, a third in 1961 and a corrected reprint of the third in 1966, Oxford: University Press.

 

 

Jeffreys used the phrase “theory of probability” in a unique way—to refer to a theory of inductive inference founded on the principle of inverse probability. The Probability is largely a treatise on theoretical statistics, though it begins with the foundations of probability and covers a range of applications comparable to that in Fisher’s Statistical Methods for Research Workers. Jeffreys was impressed by the solutions Fisher produced. In May 1937 he told Fisher of his impression that “only once in a blue moon” would we disagree about the “inference to be drawn in any particular case.” Bennett (p. 162). The trouble with Fisher’s inferences was that they lacked proper foundations!

 

The scope of the book

In all its versions the Probability was a book of 8 chapters and appendices. The titles of the chapters and their general scope remained unchanged.

 

I Fundamental notions explains how induction rests on probability and presents axioms for probability. This is the chapter with the strongest links with Scientific Inference although the treatment is much more elaborate. Whitehead and Russell’s Principia Mathematica is a powerful presence.

 

II Direct probabilities presents basic distribution theory, including the standard distributions and transform techniques. The term “direct” contrasts with “inverse.” The former indicates concern with inference from “laws” to observations, the latter with the converse.

 

III Estimation problems are ones “where we are given the form of the law, in which certain parameters can be treated as unknown, no special consideration needing to be given to any particular values, and we want the probability distribution of these parameters given the observations.” (1939, p. 94). The statement (p. 96), “Our first problem is to find a way of saying that the magnitude of a parameter is unknown, when none of the possible values need special attention” leads into a discussion of ignorance priors. Jeffreys then went through some standard problems.

 

IV Approximate methods and simplifications covers maximum likelihood, minimum χ2 and other non-Bayesian arguments. Jeffreys treats them in a remarkably positive spirit.

 

V Significance tests: one new parameter. In significance testing “our problem is to compare a suggested value of a new parameter, often 0, with the aggregate of other possible values.’’ (1939, p. 193) Jeffreys was the first writer in the inverse probability tradition to write in detail about testing. In the preface he (p. v) complained, “Modern statisticians ... for the most part have rejected the notion of the probability of a hypothesis, and thereby deprived themselves of any way of saying precisely what they mean when they decide between hypotheses.”

 

VI Significance tests: various complications treats various situations do not conform to the one new parameter format.

 

VII Frequency definitions & direct methods treats frequency definitions of probability—Venn, von Mises etc.—and the “direct methods” (frequentist methods) of Karl Pearson, Fisher and Neyman-Pearson.

 

VIII General questions recapitulates the main arguments and considers some broader issues. On this chapter and the preceding one Wilks (1941, p. 194) commented, “The discussion is almost entirely informal and non-mathematical and as such it must be regarded in the category of personal opinion.”  

 

Appendices Appendices came and went but the one constant was the set of Tables of K. K was Jeffreys’s symbol for what Good later called the “Bayes factor.” The introductory material explains the use of K and discusses the relationship between inferences based on K and on conventional tail areas—the “P integral”, as Jeffreys called it.

 

In its final form—the corrected impression (1966) of the third edition—the Probability had 459 pages compared to the first edition’s 380. The growth of the book can be seen from the table of contents of successive editions.

 

Chapters\Editions

3rd 

2nd

1st 

I. Fundamental Notions

p. 1

p. 1

p. 1

II. Direct Probabilities

57

47

46

III. Estimation  Problems

117

99

94

IV. Approximate Methods and Simplifications

193

168

145

V. Significance Tests: One New Parameter

245

220

193

VI. Significance Tests: Various Complications

332

305

268

VII. Frequency Definitions and Direct Methods

369

341

300

VIII. General  Questions

401

372

332

Appendices

425

396

356

Index

455

408

377

 

A detailed table of contents of the third edition has been prepared by Peter M Lee.

 

Jeffreys wrote articles but most of their substance went into the big book. Thus overall assessments of his work like those in the festschrift are essentially assessments of the Probability. Zellner and Kass discuss Savage’s way of classifying Jeffreys. Lindley reflects on the book and Aldrich (2005) describes the first edition against the background of Jeffreys’s intellectual development. Aldrich (2002) mentions how Jeffreys translated some of Fisher’s ideas into Bayesian terms.

  • A. Zellner (1982) Is Jeffreys a “Necessarist”?,  American Statistician, 36, 28-30 (JSTOR) with comment by Kass 390-1 (JSTOR) and reply by Zellner 392-3 (JSTOR) )

·        D. V. Lindley (1986) On Re-reading Jeffreys, pp. 35-46 of I. S. Francis et al (eds) Pacific Statistical Congress, New York: Elsevier.

·        J. Aldrich (2002) How Likelihood and Identification went Bayesian, International Statistical Review, 70, 79-98.

 

For wider perspectives see

·        Jim Berger Could Fisher, Jeffreys and Neyman Have Agreed Upon Testing? This Fisher memorial lecture has been published (with discussion) in Statistical Science, 18, 1-32 (2003) Euclid  JSTOR.

 

The next sections give some details about the three editions and how they were received. Of the three editions the first was the most important for presenting the main ideas. The second edition introduced Jeffreys’s rule the topic in the Probability which has attracted most attention. The third edition appeared at the most propitious time.

 

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The first edition 1939   back to contents

In 1939 Jeffreys was up-to-date with the work of the English statistical school, familiar with the work of Karl Pearson, Student, Fisher and Neyman-Pearson. Like most other English writers, he did not know the recent Continental work on probability, notably Kolmogorov’s Grundbegriffe. He knew only H. Cramér’s Random Variables and Probability Distributions (1937) which he recommended for a rigorous treatment of the central limit theorem.

 

In the preface (reproduced in the current edition) Jeffreys thanks Fisher, Wishart and the philosopher Richard Braithwaite for their help. None agreed with him on probability. In the text he mentions the Cambridge mathematicians M. H. A. Newman and J. E. Littlewood. It is clear, though, that after the collaboration with Wrinch came to an end Jeffreys was very much on his own in the probability project.

 

The reviews

The first edition of the Probability was reviewed in the leading journals by prominent statisticians; Irwin had worked with both Karl Pearson and Fisher while Neyman (1894-1981) (ASA  St Andrews) and Wilks (1906-64) (ASA.) would both be leaders of post-war statistics in the USA. Curiously the review in Nature was not written by a physicist but by the statistician Maurice Kendall. The reviewers respected Jeffreys’s scientific credentials but were sceptical of his system.

·        M. G. Kendall (1940) Theory of Probability by Harold Jeffreys, Nature, 145, 607-608.

·        J. Neyman (1940) Review of Theory of Probability by Harold Jeffreys, Journal of the American Statistical Association, 35, 558-559. (available through JSTOR)

·        J. O. Irwin (1941) Theory of Probability by Harold Jeffreys, Journal of the Royal Statistical Society, 104, 59-64. (available through JSTOR)

·        S. S. Wilks (1941) Theory of Probability by Harold Jeffreys, Biometrika, 32, 192-194. (available through JSTOR)

 

As well as the reviews by statisticians there were reviews by mathematicians and philosophers.

  • J. L. Doob Mathematical Reviews, (available through MathSciNet)
  • Edward L. Dodd Bulletin of the Mathematical Society, 46, (9, 1940), 739-741. (available through BAMS)  
  • Arthur H. Copeland Science New Series, 92, (Nov., 1940), 479-480. (available through JSTOR)
  • W. D. Baten National Mathematics Magazine, 15, (Dec., 1940), 159 (available through JSTOR)
  • B. O. Koopman Journal of Symbolic Logic, 8, (Mar., 1943), 34-35. (available through JSTOR)
  • W. M. M. Philosophy of Science, 7, (Apr., 1940), 263-264.  (available through JSTOR)
  • Ernest Nagel Journal of Philosophy, 37, (Sep., 1940), 524-528. (available through JSTOR)
  • R. B. Braithwaite Mind (New Series), 50, (Apr., 1941), 198-201. (available through JSTOR)

Jeffreys was a well-known physicist and his book appeared in the International Series of Monographs on Physics but it appears to have gone unnoticed by physicists.

As far as statistics was concerned, Wilks’s words seemed set to be prophetic: “It is doubtful that there will be many scholars thoroughly familiar with the system of statistical thought initiated by R. A. Fisher and extended by J. Neyman, E. S. Pearson, A Wald and others who will abandon this system in favour of the one proposed by Jeffreys in which inverse probability plays the central role.” The Annals of Mathematical Statistics the leading theoretical journal of the immediate post-war era carried no articles extending Jeffreys’s Bayesian theory before the 1960s.

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The second edition 1948       back to contents

The flow of articles on statistics abated after the publication of the Probability and in 1940-5 Jeffreys published only three articles on statistics. In 1946, however, came a major development, the introduction of the Jeffreys prior as it is usually called.

  • H. Jeffreys (1946) An Invariant Form for the Prior Probability in Estimation Problems, Proceedings of the Royal Society, A, 186, 453-461. (available through JSTOR)

 

In 1948 Jeffreys brought out a second edition of the Probability. He was now Plumian Professor of Astronomy in succession to Arthur Eddington who had taught him the theory of errors as an undergraduate. The main change in the book was an account of the invariant prior: see the preface. Jeffreys had been interested in invariance for a long time. In Scientific Inference he had discussed the consistent assignment of priors to the standard deviation, σ, and the modulus, h = 1/σ√2 , a measure of precision often used in the theory of errors.

 

In 1939 Jeffreys had sought to reconstruct modern statistics on the probability foundations he first proposed in 1919. After 1939 he did not follow the literature, nor was he impressed with the way the literature treated him. In the preface to the second edition he wrote, “I have not attempted to answer explicitly the criticisms made by reviewers, because on examination I found that they were all dealt with in the book already.”

 

The reviews

The second edition was not widely reviewed. Robbins and David were respectful but sceptical. Robbins noted that Jeffreys did not discuss the costs of making errors while David found the approach necessarily subjective. Neither was interested enough in Jeffreys’s project to comment on the changes in the new edition.

  • Herbert Robbins (1949) Review of Theory of Probability by Harold Jeffreys, Journal of the American Statistical Association, 44, No. 247 (Sep., 1949), 453.  (available through JSTOR)
  • F. N. David (1949) Review of Theory of Probability by Harold Jeffreys,  Biometrika, 36, No. 1/2 (Jun., 1949), 236.  (available through JSTOR)

 

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The third edition 1961 and the Bayesian revival   back to contents

Between 1948 and 1960 Jeffreys published very little on probability: reviews of philosophical works by Reichenbach, Russell and Carnap, an account of the “present position” and a note on the exponential family in 1960; see bibliography.  The Russell review was the most elaborate of the reviews and the BJPS article described the present position as it was in the second edition of the Probability.

·       Harold Jeffreys (1950) Bertrand Russell on Probability, Mind, 59, 313-319 (available through JSTOR).

·       Harold Jeffreys (1955) The Present Position in Probability Theory, British Journal for the Philosophy of Science, 5,  275-289 (available through JSTOR)

In 1961 Jeffreys produced a third edition of the Probability. He had retired from his chair but he went on working until his death in 1989. This edition added material to the first chapter and some mathematical appendices. The only significant new reference was to Carnap’s Logical Foundations of Probability which he reviewed in 1952. In the corrected impression of 1966 material on time series analysis was added.

 

In its third edition the Probability remained a conversation with Fisher and Principia Mathematica. Raiffa and Schlaifer’s Applied Statistical Decision Theory was published in the same year. That work reflected a huge change in the statistical climate as more statisticians became interested in Bayesian ideas. There were many publications that contributed to the Bayesian revival. In Britain Good (ASA conversation St Andrews) was influenced by Jeffreys and by Cambridge probability more generally; like Jeffreys and unlike most statisticians, Good was interested in logic and the philosophy of science. The more influential US line of Savage and Schlaifer was very different. Their foundations added to the “personalism” (subjective probability) of Ramsey and de Finetti the “behavioralism” (decision-orientation) of Neyman and Wald; the expected utility theory of von Neumann and Morgenstern was also a great influence. Jeffreys was known to the American statisticians but was not much of an influence. For Savage (1954, p. 276) the Probability was “an ingenious and vigorous defense of a necessary view, similar to, but more sophisticated than Laplace’s.” However, Savage did not discuss this defence: his representative “necessarists” (p. 61) were Keynes and Carnap. Raiffa and Schlaifer (1961, p. ix) include Jeffreys in their general debt—to “Neyman, Pearson, Jeffreys, Von Neumann, Wald, Blackwell, Girshick and Savage.” 

  • I. J. Good (1950) Probability and the Weighing of Evidence, London: Griffin.
  • L. J. Savage (1954/72) The Foundations of Statistics. Expanded second edition published by Dover (New York) of a book originally published by Wiley (New York).
  • Howard Raiffa and Robert Schlaifer (1961) Applied Statistical Decision Theory, Cambridge, Mass.: MIT Press.

 

Jeffreys seems to have paid no attention to these developments in the statistical literature; his expected utility was the moral expectation of Laplace. In 1963 when he reviewed the proceedings of a conference on the new work he focussed on the probability side.

  • Harold Jeffreys (1963) Review of The Foundations of Statistical Inference by L. J. Savage and others, Technometrics, 5, 407-410. (available through JSTOR)

 

The reviews

The third edition, like the second, was not widely reviewed. However, for the first time there were Jeffreys enthusiasts to receive it.

  • I. J. Good (1962) Theory of Probability. by Harold Jeffreys, Journal of the Royal Statistical Society A,125, 487-48.  (available through JSTOR) 

·       D. V. Lindley (1962), Theory of Probability. by Harold Jeffreys, Journal of the American Statistical Association, 57, 922-924 (available through JSTOR:)

·       F. Zitek Mathematical Reviews, (available through MathSciNet)

·       M. Arcones (1998 reprint) Mathematical Reviews, (available through MathSciNet)

 

Good and Lindley wrote in very different terms from earlier reviewers. Lindley’s review begins

 

This is probably the most original and important book in statistics that has appeared in the last forty years. The only serious competitor is Fisher’s “Statistical Methods for Research Workers” [of 1925]. The distinction between the two is that Fisher is usually right for the wrong reasons, whereas Jeffreys gets the reasoning broadly correct, as well as the answers.

 

Good has some philosophical remarks on how Jeffreys failed to notice other writers, including “Pioneers often ignore the work of those who have stood on their shoulders” and “if Jeffreys had tried to bring the book up to date, rather than to revise it in part, we might have had much longer to wait for the present edition.” 

 

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Afterwards  back to contents

Jeffreys’s influence on Savage and Schlaifer was not great but soon there were works that did reflect his influence, including           

  • A. Zellner (1971) An Introduction to Bayesian Inference in Econometrics. New York: Wiley.
  • G. E. P. Box & G. C. Tiao (1973) Bayesian Inference in Statistical Analysis, Reading, MA: Addison-Wesley

Apart from Arnold Zellner ET interview the most forceful advocate of Jeffreys’s ideas has been the physicist E. T. Jaynes (1922-98). His magnum opus is

  • E. T. Jaynes (2003) Probability: the Logic of Science edited by G. Larry Bretthorst. Cambridge University Press. Sample material is available from the publisher. A fragmentary edition (1994) is available on the web Probability: the Logic of Science

 

Of Jeffreys’s contributions the one that found most application and criticism was his rule for constructing uninformative priors. Invariance and the paradoxes associated with the improper priors that application of his rule generates are discussed by Hartigan and Dawid et al. The review by Kass & Wasserman follows the evolution of Jeffreys’s thought as well as considering later work

  • J. Hartigan (1964) Invariant Prior Distributions, Annals of Mathematical Statistics, 35, 836-845. (available through JSTOR)

·        P. Dawid, M. Stone & J. V. Zidek (1973) Marginalization Paradoxes in Bayesian and Structural Inference, (with discussion) Journal of the Royal Statistical Society. B, 35, 189-233. (available through JSTOR)

·        Robert E. Kass & Larry Wasserman (1996) The Selection of Prior Distributions by Formal Rules, Journal of the American Statistical Association, 91, 1343-1370. (available through JSTOR)

 

Seventy years after the publication of the first edition of Theory of Probability the journal Statistical Science had a symposium

  • Christian P. Robert, Nicolas Chopin and Judith Rousseau (2009) Harold Jeffreys’s Theory of Probability Revisited, Statistical Science, 24, (2), 141-172.
  • Comments by José M. Bernardo, 173-175; Andrew Gelman, 176-178; Robert Kass, 179-182; Dennis Lindley, 183-184; Stephen Senn, 185-186; Arnold Zellner, 187-190.
  • Rejoinder by Christian P. Robert, Nicolas Chopin and Judith Rousseau, 191-194.

 

 

 

Other links

·        The International Society for Bayesian Analysis (ISBA)

·        Peter Lee’s History of Statistics  has many useful links.

·        My Figures from the History of Probability and Statistics sketches the history of probability and statistics and has notes on some key people. 

 

 

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