Likelihood
and Probability in R. A. Fisher’s *Statistical
Methods for Research Workers*

There is a passage in
Chapter 1 of Fisher’s *Statistical Methods
for Research Workers* describing the proper roles of probability and
likelihood. It criticises the Bayesian misuse of probability (termed “inverse
probability”) and seems to anticipate the 1960s discussion of the likelihood
principle. (See the entries *Bayesian*,
*Likelihood* and *Likelihood Principle* in *Probability
& Statistics on the Earliest Uses Pages.)* The purpose of this presentation is
to make the passage available and, more especially, to show the changes Fisher
made in the later editions of the book.

The book was first published in 1925 and the division
of responsibility between probability and likelihood, which it lays down,* *follows what Fisher had written in On the "Probable Error" of a Coefficient of
Correlation Deduced from a Small Sample
(1921). This paper not only describes the division (p. 24), it shows how
to do likelihood inference, or at least how to construct likelihood intervals
(see p. 25). However the wide-ranging and masterly On the
Mathematical Foundations of Theoretical Statistics (1922) was *not*
a likelihood text in the same sense. It also presents maximum likelihood but
the properties it emphasises are properties in ‘repeated sampling.’ The Theory
of Statistical Estimation (1925) takes the
same approach. Two New
Properties of Mathematical Likelihood (1934)
added a conditional dimension to Fisher’s theory of estimation but the concern was still with repeated sampling
properties. Although likelihood inference was proclaimed as a
fundamental form of inference in all editions of *Statistical Methods for Research Workers*, it was not
developed any further for more than 30 years. Likelihood inference re-appears
in *Statistical Methods and Scientific
Inference* (1956) and the discussion there picks up from where the
1921 paper and *Statistical Methods for
Research Workers *had left off, “The likelihood supplies a natural
order of preference among the possibilities under consideration” (chapter III,
§6).

By 1925 Fisher’s opposition to Bayesian inference (as
it would now be called) was already fixed; see his On the
Mathematical Foundations of Theoretical Statistics. Over the life-time of *Statistical Methods for Research Workers*
Fisher’s views on Bayes, the person, changed and he also debated with
Bayesians, such as Harold Jeffreys (see Harold Jeffreys
as a Statistician). However these developments led to no rewriting of the
1925 passage. The rewriting was done primarily to accommodate the *fiducial* argument, which gave probability
a role in inference—a *legitimate*
role in Fisher’s eyes, unlike the Bayesian pretence of a role. The changes
began to appear in the 1932 (4^{th}) edition, following the publication
of his Inverse
Probability which introduced the fiducial argument. The article appeared in 1930 and
Fisher slowly fiducialised the book by rewriting the
passage as well as by making changes elsewhere. He did *not* revise his account of likelihood
except to describe the power function of Neyman and Pearson as a specialised
application of it!

Most of the relevant
documents can be found on the web: Fisher’s articles from the University of
Adelaide’s R. A.
Fisher Digital Archive and the first edition of *SMRW* on Christopher Green’s site. These sites give the full
references.

*Statistical Methods for Research Workers*(The first edition is available in Christopher Green’s*Classics in the History of Psychology*__.)__-
*Statistical Methods for Research Workers*(The 14th edition is in print as part of*Statistical Methods*,*Experimental Design and Scientific Inference*, Oxford: Oxford University Press, 1990.) - On the "Probable Error" of a Coefficient of Correlation Deduced from a Small Sample. (1921)

· On the Mathematical Foundations of Theoretical Statistics. (1922)

- Theory of Statistical Estimation. (1925)
- Inverse Probability. (1930)
- Two New Properties of Mathematical Likelihood. (1934)
*Statistical Methods and Scientific Inference*, 3 editions, 1956/59/74, Edinburgh: Oliver & Boyd. (The 3rd edition is in print as part of*Statistical Methods*,*Experimental Design and Scientific Inference*, Oxford: Oxford University Press, 1990.)

George Barnard developed Fisher’s division between probability and likelihood in a 1949 paper but the work had little impact. The same cannot be said of Allan Birnbaum’s effort, which appeared when there was a strong revival of interest in foundations. Edwards’s book develops a system of statistics based on likelihood. Berger & Wolpert provide a survey.

- G. A. Barnard (1949) Statistical Inference,
*Journal of the Royal Statistical Society. Series B (Methodological)*, 11, 115-149.*JSTOR*. (For a personal appreciation of Barnard see Lindley's George. Barnard (1915-2002) ) - Barnard sent Fisher a copy of his paper and in
reply Fisher congratulated him on his “enterprise”: see pp. 5-6 of J. H. Bennett (1990) (ed)
*Statistical Inference and Analysis: Selected Correspondence of R. A. Fisher*, Oxford, University Press. - Allan Birnbaum (1962) On the Foundations of
Statistical Inference"
*Journal of the American Statistical Association,***57,**269-306 (with comments by L. J. Savage, G. A. Barnard, J. Cornfield, Irwin Bross, G. E. P. Box, I. J. Good, D. V. Lindley, C. W. Clunies-Ross, J. W. Pratt, H. Levene, T. Goldman, A. P. Dempster, O. Kempthorne and reply by Birnbaum, 307-326).*JSTOR* - Between 1958 and –60
Birnbaum sent Fisher copies
of his papers but Fisher was not interested in Birnbaum’s project of improving Neyman’s
theory. Birnbaum did not send Fisher a copy of his likelihood paper which
was presented at a meeting in December 1961. However the paper may have
been enclosed with a letter from Kempthorne to
which Fisher replied on 19
^{th}February 1962; Fisher described Birnbaum as “a very bewildered type.” In a letter to H. E. Kyburg dated 14^{th}May Fisher mentioned that “Likelihood Statements” had “recently been rediscovered by Birnbaum”: see pp. 188-9 of J. H. Bennett (1990) (ed). Birnbaum's paper appeared in the June issue of*JASA*. On 29^{th}July Fisher died. - A. W. F. Edwards (1972/1992)
*Likelihood*(the second expanded edition reprints the first with some related articles and a new preface), Baltimore: Johns Hopkins University Press. - J. O. Berger, R. L. Wolpert
(1988)
*The**Likelihood Principle*(2^{nd}. Edition), Hayward, Calif.: Institute of Mathematical Statistics.

For a perspective on Fisher’s *Statistical Methods for Research Workers* see Edwards,
(2005). Aldrich (2008) describes Fisher’s attitudes to Bayesian inference and
to Bayes the man. Edwards
(1999) provides a nice introduction to likelihood and its history. Aldrich
(1997) gives a detailed account of Fisher’s ideas on likelihood to 1922, while
Aldrich (2000) describes the development of the fiducial argument and the
consequences of this development for the likelihood/probability division. Hald
has a longer perpective. The *Guide*
provides additional information on Fisher, including references on the fiducial
argument.* Figures contains* general
information on the history of statistics.

- A. W. F. Edwards,
(2005) “R. A. Fisher, Statistical Methods for Research Workers, 1925” in
I. Grattan-Guinness (ed.)
*Landmark Writings in Western Mathematics : Case Studies, 1640-1940*, Amsterdam: Elsevier. - John Aldrich (2008) R. A. Fisher
on Bayes and Bayes’ Theorem.
*Bayesian Analysis*,**3**, 161-170. here - A. W. F. Edwards (1999) “Likelihood” preliminary version of IESBS entry.
- John Aldrich (1997) R. A. Fisher and the Making
of Maximum Likelihood 1912-22,
*Statistical Science*,**12**, 162-176. Project Euclid*JSTOR* - Anders Hald
(1999) On the History of Maximum Likelihood in Relation to Inverse
Probability and Least Squares,
*Statistical Science*,**14**, 214-222. Project Euclid*JSTOR* - John Aldrich (2000) Fisher’s “Inverse
Probability” of 1930, I
*nternational Statistical Review*,**68**, 155-172. pdf - John Aldrich
*A Guide to R. A. Fisher* - John Aldrich
__Figures from the History of Probability and Statistics__

*John Aldrich, University of
Southampton, Southampton, UK.** ***(****home****) December
2003. Most
recent changes February 2015. **