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 (4th) 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.
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.
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.
John Aldrich, University of Southampton, Southampton, UK. (home) December 2003. Most recent changes February 2015.