Frequentist probability

"Statistical probability" redirects here. For the TV series episode, see Statistical Probabilities.

John Venn, who provided a thorough exposition of frequentist probability in his book, The Logic of Chance^[1]

Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in infinitely many trials (the long-run probability).^[2] Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). The continued use of frequentist methods in scientific inference, however, has been called into question.^[3][4][5]

The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation. In the classical interpretation, probability was defined in terms of the principle of indifference, based on the natural symmetry of a problem, so, for example, the probabilities of dice games arise from the natural symmetric 6-sidedness of the cube. This classical interpretation stumbled at any statistical problem that has no natural symmetry for reasoning.

Definition

In the frequentist interpretation, probabilities are discussed only when dealing with well-defined random experiments. The set of all possible outcomes of a random experiment is called the sample space of the experiment. An event is defined as a particular subset of the sample space to be considered. For any given event, only one of two possibilities may hold: It occurs or it does not. The relative frequency of occurrence of an event, observed in a number of repetitions of the experiment, is a measure of the probability of that event. This is the core conception of probability in the frequentist interpretation.

A claim of the frequentist approach is that, as the number of trials increases, the change in the relative frequency will diminish. Hence, one can view a probability as the limiting value of the corresponding relative frequencies.

Scope

The frequentist interpretation is a philosophical approach to the definition and use of probabilities; it is one of several such approaches. It does not claim to capture all connotations of the concept 'probable' in colloquial speech of natural languages.

As an interpretation, it is not in conflict with the mathematical axiomatization of probability theory; rather, it provides guidance for how to apply mathematical probability theory to real-world situations. It offers distinct guidance in the construction and design of practical experiments, especially when contrasted with the Bayesian interpretation. As to whether this guidance is useful, or is apt to mis-interpretation, has been a source of controversy. Particularly when the frequency interpretation of probability is mistakenly assumed to be the only possible basis for frequentist inference. So, for example, a list of mis-interpretations of the meaning of p-values accompanies the article on p-values; controversies are detailed in the article on statistical hypothesis testing. The Jeffreys–Lindley paradox shows how different interpretations, applied to the same data set, can lead to different conclusions about the 'statistical significance' of a result.^{[citation needed]}

As Feller notes:^[a]

There is no place in our system for speculations concerning the probability that the sun will rise tomorrow. Before speaking of it we should have to agree on an (idealized) model which would presumably run along the lines "out of infinitely many worlds one is selected at random ..." Little imagination is required to construct such a model, but it appears both uninteresting and meaningless.^[6]

History

Main article: History of probability

The frequentist view may have been foreshadowed by Aristotle, in Rhetoric,^[7] when he wrote:

the probable is that which for the most part happens — Aristotle Rhetoric^[8]

Poisson (1837) clearly distinguished between objective and subjective probabilities.^[9] Soon thereafter a flurry of nearly simultaneous publications by Mill, Ellis (1843)^[10] and Ellis (1854),^[11] Cournot (1843),^[12] and Fries introduced the frequentist view. Venn (1866, 1876, 1888)^[1] provided a thorough exposition two decades later. These were further supported by the publications of Boole and Bertrand. By the end of the 19th century the frequentist interpretation was well established and perhaps dominant in the sciences.^[9] The following generation established the tools of classical inferential statistics (significance testing, hypothesis testing and confidence intervals) all based on frequentist probability.

Alternatively,^[13] Bernoulli^[b] understood the concept of frequentist probability and published a critical proof (the weak law of large numbers) posthumously (Bernoulli, 1713).^[14] He is also credited with some appreciation for subjective probability (prior to and without Bayes theorem).^[15][c][16] Gauss and Laplace used frequentist (and other) probability in derivations of the least squares method a century later, a generation before Poisson.^[13] Laplace considered the probabilities of testimonies, tables of mortality, judgments of tribunals, etc. which are unlikely candidates for classical probability. In this view, Poisson's contribution was his sharp criticism of the alternative "inverse" (subjective, Bayesian) probability interpretation. Any criticism by Gauss or Laplace was muted and implicit. (However, note that their later derivations of least squares did not use inverse probability.)

Major contributors to "classical" statistics in the early 20th century included Fisher, Neyman, and Pearson. Fisher contributed to most of statistics and made significance testing the core of experimental science, although he was critical of the frequentist concept of "repeated sampling from the same population";^[17] Neyman formulated confidence intervals and contributed heavily to sampling theory; Neyman and Pearson paired in the creation of hypothesis testing. All valued objectivity, so the best interpretation of probability available to them was frequentist.

All were suspicious of "inverse probability" (the available alternative) with prior probabilities chosen by using the principle of indifference. Fisher said, "... the theory of inverse probability is founded upon an error, [referring to Bayes theorem] and must be wholly rejected."^[18] While Neyman was a pure frequentist,^[19][d] Fisher's views of probability were unique: Both Fisher and Neyman had nuanced view of probability. von Mises offered a combination of mathematical and philosophical support for frequentism in the era.^[20][21]

Etymology

According to the Oxford English Dictionary, the term frequentist was first used by M.G. Kendall in 1949, to contrast with Bayesians, whom he called non-frequentists.^[22][23] Kendall observed

3. ... we may broadly distinguish two main attitudes. One takes probability as 'a degree of rational belief', or some similar idea...the second defines probability in terms of frequencies of occurrence of events, or by relative proportions in 'populations' or 'collectives';^{[23](p 101)}

...

12. It might be thought that the differences between the frequentists and the non-frequentists (if I may call them such) are largely due to the differences of the domains which they purport to cover.^{[23](p 104)}

...

I assert that this is not so ... The essential distinction between the frequentists and the non-frequentists is, I think, that the former, in an effort to avoid anything savouring of matters of opinion, seek to define probability in terms of the objective properties of a population, real or hypothetical, whereas the latter do not. [emphasis in original]

"The Frequency Theory of Probability" was used a generation earlier as a chapter title in Keynes (1921).^[7]

The historical sequence:

1. Probability concepts were introduced and much of the mathematics of probability derived (prior to the 20th century)
2. classical statistical inference methods were developed
3. the mathematical foundations of probability were solidified and current terminology was introduced (all in the 20th century).

The primary historical sources in probability and statistics did not use the current terminology of classical, subjective (Bayesian), and frequentist probability.

Alternative views

Main article: Probability interpretations

Probability theory is a branch of mathematics. While its roots reach centuries into the past, it reached maturity with the axioms of Andrey Kolmogorov in 1933. The theory focuses on the valid operations on probability values rather than on the initial assignment of values; the mathematics is largely independent of any interpretation of probability.

Applications and interpretations of probability are considered by philosophy, the sciences and statistics. All are interested in the extraction of knowledge from observations—inductive reasoning. There are a variety of competing interpretations;^[24] All have problems. The frequentist interpretation does resolve difficulties with the classical interpretation, such as any problem where the natural symmetry of outcomes is not known. It does not address other issues, such as the dutch book.

• Classical probability assigns probabilities based on physical idealized symmetry (dice, coins, cards). The classical definition is at risk of circularity: Probabilities are defined by assuming equality of probabilities.^[25] In the absence of symmetry the utility of the definition is limited.

• Subjective (Bayesian) probability (a family of competing interpretations) considers degrees of belief: All practical "subjective" probability interpretations are so constrained to rationality as to avoid most subjectivity. Real subjectivity is repellent to some definitions of science which strive for results independent of the observer and analyst.^{[citation needed]} Other applications of Bayesianism in science (e.g. logical Bayesianism) embrace the inherent subjectivity of many scientific studies and objects and use Bayesian reasoning to place boundaries and context on the influence of subjectivities on all analysis.^[26] The historical roots of this concept extended to such non-numeric applications as legal evidence.

• Propensity probability views probability as a causative phenomenon rather than a purely descriptive or subjective one.^[24]

Footnotes

1. Feller's comment is a criticism of Pierre-Simon Laplace's solution to the "tomorrow's sunrise" problem that used an alternative probability interpretation.
   Despite Laplace's explicit and immediate disclaimer in the source, based on Laplace's personal expertise in both astronomy and probability, two centuries of nattering criticism have followed.
2. The Swiss mathematician Jacob Bernoulli of the famous Bernoulli family lived in a multi-lingual country and himself had regular correspondance and contacts with speakers of German and French, and published in Latin – all of which he spoke fluently. He comfortably and frequently used the three names "Jacob", "James", and "Jacques", depending on the language he was speaking or writing.
3. Bernoulli provided a classical example of drawing many black and white pebbles from an urn (with replacement). The sample ratio allowed Bernoulli to infer the ratio in the urn, with tighter bounds as the number of samples increased.
   Historians can interpret the example as classical, frequentist, or subjective probability. David writes, "James has definitely started here the controversy on inverse probability ..." Bernoulli wrote generations before Bayes, LaPlace and Gauss. The controversy continues. — David (1962), pp.  137–138^[16]
4. Jerzy Neyman's derivation of confidence intervals embraced the measure theoretic axioms of probability published by Andrey Kolmogorov a few years earlier, and referenced the 'subjective probability (Bayesian) definitions that Jeffreys had published earlier in the decade. Neyman defined frequentist probability (under the name classical) and stated the need for randomness in the repeated samples or trials. He accepted in principle the possibility of multiple competing theories of probability, while expressing several specific reservations about the existing alternative probability interpretation.^[19]

Citations

1. Venn, John (1888) [1866, 1876]. The Logic of Chance (3rd ed.). London, UK: Macmillan & Co. – via Internet Archive (archive.org. An essay on the foundations and province of the theory of probability, with especial reference to its logical bearings and its application to moral and social science, and to statistics.
2. Kaplan, D. (2014). Bayesian Statistics for the Social Sciences. Methodology in the Social Sciences. Guilford Publications. p. 4. ISBN 978-1-4625-1667-4. Retrieved 23 April 2022.
3. Goodman, Steven N. (1999). "Toward evidence-based medical statistics. 1: The p value fallacy". Annals of Internal Medicine. 130 (12): 995–1004. doi:10.7326/0003-4819-130-12-199906150-00008. PMID 10383371. S2CID 7534212.
4. Morey, Richard D.; Hoekstra, Rink; Rouder, Jeffrey N.; Lee, Michael D.; Wagenmakers, Eric-Jan (2016). "The fallacy of placing confidence in confidence intervals". Psychonomic Bulletin & Review. 23 (1): 103–123. doi:10.3758/s13423-015-0947-8. PMC 4742505. PMID 26450628.
5. Matthews, Robert (2021). "The p-value statement, five years on". Significance. 18 (2): 16–19. doi:10.1111/1740-9713.01505. S2CID 233534109.
6. Feller, W. (1957). An Introduction to Probability Theory and Its Applications. Vol. 1. p. 4.
7. Keynes, J.M. (1921). "Chapter VIII – The frequency theory of probability". A Treatise on Probability.
8. Aristotle. Rhetoric. Bk 1, Ch 2.

   discussed in

   Franklin, J. (2001). The Science of Conjecture: Evidence and probability before Pascal. Baltimore, MD: The Johns Hopkins University Press. p. 110. ISBN 0801865697.
9. Gigerenzer, Gerd; Swijtink, Porter; Daston, Beatty; Daston, Krüger (1989). The Empire of Chance : How probability changed science and everyday life. Cambridge, UK / New York, NY: Cambridge University Press. pp. 35–36, 45. ISBN 978-0-521-39838-1.
10. Ellis, R.L. (1843). "On the foundations of the theory of probabilities". Transactions of the Cambridge Philosophical Society. 8.
11. Ellis, R.L. (1854). "Remarks on the fundamental principles of the theory of probabilities". Transactions of the Cambridge Philosophical Society. 9.
12. Cournot, A.A. (1843). Exposition de la théorie des chances et des probabilités. Paris, FR: L. Hachette – via Internet Archive (archive.org).
13. Hald, Anders (2004). A history of Parametric Statistical Inference from Bernoulli to Fisher, 1713 to 1935. København, DM: Anders Hald, Department of Applied Mathematics and Statistics, University of Copenhagen. pp. 1–5. ISBN 978-87-7834-628-5.
14. Bernoulli, Jakob (1713). Ars Conjectandi: Usum & applicationem praecedentis doctrinae in civilibus, moralibus, & oeconomicis [The Art of Conjecture: The use and application of previous experience in civil, moral, and economic topics] (in Latin).
15. Fienberg, Stephen E. (1992). "A Brief History of Statistics in Three and One-half Chapters: A Review Essay". Statistical Science. 7 (2): 208–225. doi:10.1214/ss/1177011360.
16. David, F.N. (1962). Games, Gods, & Gambling. New York, NY: Hafner. pp. 137–138.
17. Rubin, M. (2020). ""Repeated sampling from the same population?" A critique of Neyman and Pearson's responses to Fisher". European Journal for Philosophy of Science. 10 (42): 1–15. doi:10.1007/s13194-020-00309-6. S2CID 221939887.
18. Fisher, R.A. Statistical Methods for Research Workers.
19. Neyman, Jerzy (30 August 1937). "Outline of a theory of statistical estimation based on the classical theory of probability". Philosophical Transactions of the Royal Society of London A. 236 (767): 333–380. Bibcode:1937RSPTA.236..333N. doi:10.1098/rsta.1937.0005.
20. von Mises, Richard (1981) [1939]. Probability, Statistics, and Truth (in German and English) (2nd, rev. ed.). Dover Publications. p. 14. ISBN 0486242145.
21. Gilles, Donald (2000). "Chapter 5 – The frequency theory". Philosophical Theories of Probability. Psychology Press. p. 88. ISBN 9780415182751.
22. "Earliest known uses of some of the words of probability & statistics". leidenuniv.nl. Leidin, NL: Leiden University.
23. Kendall, M.G. (1949). "On the Reconciliation of Theories of Probability". Biometrika. 36 (1–2): 101–116. doi:10.1093/biomet/36.1-2.101. JSTOR 2332534. PMID 18132087.
24. Hájek, Alan (21 October 2002). "Interpretations of probability". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy – via plato.stanford.edu.
25. Ash, Robert B. (1970). Basic Probability Theory. New York, NY: Wiley. pp. 1–2.
26. Fairfield, Tasha; Charman, Andrew E. (15 May 2017). "Explicit Bayesian analysis for process tracing: Guidelines, opportunities, and caveats". Political Analysis. 25 (3): 363–380. doi:10.1017/pan.2017.14. S2CID 8862619.

References

• Bridgman, P.W. (1927). The Logic of Modern Physics.

• Church, Alonzo (1940). The Concept of a Random Sequence.

• Cramér, Harald (1946). Mathematical Methods of Statistics.

• Feller, William (1957). An Introduction to Probability Theory and its Applications.

• Martin-Löf, P. (1966). On the Concept of a Random Sequence.

• von Mises, Richard (1939) [1928 (in German)]. Probability, Statistics, and Truth.

• Neyman, Jerzy (1950). First Course in Probability and Statistics.

• Reichenbach, Hans (1949) [1935 (in German)]. The Theory of Probability.

• Russell, Bertrand (1948). Human Knowledge.

• Friedman, C. (1999). "The frequency interpretation in probability". Advances in Applied Mathematics. 23 (3): 234–254. doi:10.1006/aama.1999.0653. "alt. source" (PS). utexas.edu/~friedman. Austin, TX: University of Texas.