Editing Frequentist probability (section)

== History ==
{{main|History of probability}}
The frequentist view may have been foreshadowed by [[Aristotle]], in ''[[Rhetoric (Aristotle)|Rhetoric]]'',<ref name=keynesVIII>
{{cite book
 |last = Keynes |first = J.M. |author-link = John Maynard Keynes
 |title = A Treatise on Probability
 |year = 1921
 |chapter = Chapter&nbsp;VIII – The frequency theory of probability
}}
</ref>
when he wrote:
{{Quote|
the probable is that which for the most part happens — [[Aristotle]] [[Rhetoric (Aristotle)|''Rhetoric'']]<ref name=aristorhetor>
{{cite book
 |author = [[Aristotle]]
 |title = Rhetoric  |title-link = Rhetoric (Aristotle) 
 |at=Bk&nbsp;1, Ch&nbsp;2
}}
: discussed in
{{cite book
 |first = J. |last = Franklin
 |title = The Science of Conjecture: Evidence and probability before Pascal
 |year = 2001
 |publisher = The Johns Hopkins University Press
 |place = Baltimore, MD
 |ISBN = 0801865697
 |page = 110
}}
</ref>
}}

[[Siméon Denis Poisson|Poisson]] (1837) clearly distinguished between objective and subjective probabilities.<ref name=Gig89>
{{cite book
 |last1 = Gigerenzer |first1 = Gerd             |last2 = Swijtink |first2 = Porter
 |last3 = Daston     |first3 = Beatty           |last4 = Daston   |first4 = Krüger
 |year = 1989
 |title = The Empire of Chance : How probability changed science and everyday life
 |publisher = Cambridge University Press
 |isbn = 978-0-521-39838-1
 |location = Cambridge, UK / New York, NY
 |pages = 35–36, 45
}}
</ref>
Soon thereafter a flurry of nearly simultaneous publications by [[John Stuart Mill|Mill]], [[Robert Leslie Ellis|Ellis]] (1843)<ref name=ellisfound>
{{cite journal
 |last = Ellis  |first = R.L. |author-link = Robert Leslie Ellis
 |year = 1843
 |title = On the foundations of the theory of probabilities
 |journal = [[Transactions of the Cambridge Philosophical Society]]
 |volume = 8
}}
</ref>
and Ellis (1854),<ref name=ellisfund>
{{cite journal
 |last = Ellis |first = R.L. |author-link = Robert Leslie Ellis
 |year = 1854
 |title = Remarks on the fundamental principles of the theory of probabilities
 |journal = [[Transactions of the Cambridge Philosophical Society]]
 |volume = 9
}}
</ref> [[Antoine Augustin Cournot|Cournot]] (1843),<ref>
{{cite book
 |last = Cournot |first = A.A. |author-link = Antoine Augustin Cournot
 |year = 1843
 |title = Exposition de la théorie des chances et des probabilités
 |publisher = L. Hachette
 |place = Paris, FR
 |url = https://archive.org/details/expositiondelat00courgoog 
 |via = [[Internet Archive]] (archive.org)
}}
</ref>
and [[Jakob Friedrich Fries|Fries]] introduced the frequentist view. [[John Venn|Venn]] (1866, 1876, 1888)<ref name=Venn-1888/> provided a thorough exposition two decades later.
These were further supported by the publications of [[George Boole|Boole]] and [[Joseph Louis François Bertrand|Bertrand]]. By the end of the 19th&nbsp;century the frequentist interpretation was well established and perhaps dominant in the sciences.<ref name=Gig89/> The following generation established the tools of classical inferential statistics (significance testing, hypothesis testing and confidence intervals) all based on frequentist probability.

Alternatively,<ref name=Anders-2004/>
[[Jacob Bernoulli|Bernoulli]]{{efn|
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.
}}
understood the concept of frequentist probability and published a critical proof (the [[weak law of large numbers]]) posthumously (Bernoulli, 1713).<ref>{{cite book
 |first = Jakob  |last = Bernoulli
 |year=1713
 |title = Ars Conjectandi: Usum & applicationem praecedentis doctrinae in civilibus, moralibus, & oeconomicis
 |lang=la
 |trans-title = The Art of Conjecture: The use and application of previous experience in civil, moral, and economic topics
}}
</ref>
He is also credited with some appreciation for subjective probability (prior to and without [[Bayes' theorem]]).<ref>
{{cite journal
 | last = Fienberg | first = Stephen E.
 | year = 1992
 | title = A Brief History of Statistics in Three and One-half Chapters: A Review Essay
 | journal = Statistical Science
 | volume = 7 | number = 2 | pages = 208–225
 | doi=10.1214/ss/1177011360| doi-access = free
}}
</ref>{{efn|
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, ''"[[Jacob Bernoulli|James]] has definitely started here the controversy on inverse probability&nbsp;..."'' Bernoulli wrote generations before Bayes, LaPlace and Gauss. The controversy continues. — {{harvp|David|1962|pp= 137–138}}<ref name=David-1962/>
}}<ref name=David-1962>
{{cite book
 | last = David | first = F.N.
 | year = 1962
 | title = Games, Gods, & Gambling
 | location = New York, NY
 | publisher = Hafner
 | pages = 137–138
}}
</ref>
[[Carl Friedrich Gauss|Gauss]] and [[Pierre-Simon Laplace|Laplace]] used frequentist (and other) probability in derivations of the least squares method a century later, a generation before Poisson.<ref name=Anders-2004>
{{cite book
 | last = Hald | first = Anders
 | year = 2004
 | title = A history of Parametric Statistical Inference from Bernoulli to Fisher, 1713 to 1935
 | publisher = Anders Hald, Department of Applied Mathematics and Statistics, [[University of Copenhagen]]
 | location = København, DM
 | isbn = 978-87-7834-628-5
 | pages = 1–5
}}
</ref>
[[Pierre-Simon Laplace|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 [[Carl Friedrich Gauss|Gauss]] or [[Pierre-Simon Laplace|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 [[Ronald Aylmer Fisher|Fisher]], [[Jerzy Neyman|Neyman]], and [[Egon Pearson|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"'';<ref>
{{cite journal
 |last = Rubin |first = M.
 |year=2020
 |title="Repeated sampling from the same population?" A critique of Neyman and Pearson's responses to Fisher
 |journal=European Journal for Philosophy of Science
 |volume=10 |issue=42 |pages=1–15
 |doi=10.1007/s13194-020-00309-6
 |s2cid=221939887
 |url=https://doi.org/10.1007/s13194-020-00309-6
}}
</ref>
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, ''"...&nbsp;the theory of inverse probability is founded upon an error, [referring to Bayes' theorem] and must be wholly rejected."''<ref>
{{cite book
 |first = R.A. |last = Fisher  |author-link = Ronald Aylmer Fisher
 |title = Statistical Methods for Research Workers
}}
</ref>
While Neyman was a pure frequentist,<ref name=Neyman-1937>
{{cite journal
 |last = Neyman |first = Jerzy  |author-link = Jerzy Neyman
 |date = 30 August 1937
 |title = Outline of a theory of statistical estimation based on the classical theory of probability
 |journal = [[Philosophical Transactions of the Royal Society of London]] A
 |volume = 236  |issue = 767  |pages = 333–380
 |bibcode = 1937RSPTA.236..333N
 |doi = 10.1098/rsta.1937.0005 |doi-access=free
}}
</ref>{{efn|
[[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 [[Sir Harold Jeffreys|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.<ref name=Neyman-1937/>
}}
Fisher's views of probability were unique: Both Fisher and Neyman had nuanced view of probability. [[Richard von Mises|von Mises]] offered a combination of mathematical and philosophical support for frequentism in the era.<ref>
{{cite book
 |last = von Mises |first = Richard  |author-link = Richard von Mises
 |orig-year = 1939  |year = 1981
 |title = Probability, Statistics, and Truth  |edition = 2nd, rev.
 |lang = de, en
 |publisher = Dover Publications
 |ISBN = 0486242145
 |page = 14 
}}
</ref><ref>
{{cite book
 |first = Donald  |last = Gilles
 |year = 2000
 |chapter=Chapter&nbsp;5 – The frequency theory
 |title = Philosophical Theories of Probability
 |publisher = Psychology Press
 |ISBN = 9780415182751
 |page = 88
}}
</ref>