Editing Bayes' theorem (section)

== History ==

Bayes' theorem is named after [[Thomas Bayes]] ({{IPAc-en|b|eɪ|z}}), a minister, statistician, and philosopher. Bayes used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter. His work was published in 1763 as ''[[An Essay Towards Solving a Problem in the Doctrine of Chances]]''. Bayes studied how to compute a distribution for the probability parameter of a [[binomial distribution]] (in modern terminology). After Bayes's death, his family gave his papers to a friend, the minister, philosopher, and mathematician [[Richard Price]].

Price significantly edited the unpublished manuscript for two years before sending it to a friend who read it aloud at the [[Royal Society]] on 23 December 1763.<ref name="Liberty's Apostle">{{cite book |last1=Frame |first1=Paul |url=https://www.uwp.co.uk/book/libertys-apostle-richard-price-his-life-and-times/ |title=Liberty's Apostle |date=2015 |publisher=University of Wales Press |isbn=978-1783162161 |location=Wales |pages=44 |language=en |access-date=23 February 2021}}</ref> Price edited<ref>{{cite book |first = Richard |last = Allen |title=David Hartley on Human Nature |url = https://books.google.com/books?id=NCu6HhGlAB8C&pg=PA243 |access-date=16 June 2013 |year=1999 |publisher=SUNY Press |isbn=978-0791494516 |pages=243–244}}</ref> Bayes's major work "An Essay Towards Solving a Problem in the Doctrine of Chances" (1763), which appeared in ''[[Philosophical Transactions]]'',<ref name="Price1763">{{cite journal |doi=10.1098/rstl.1763.0053 |journal=Philosophical Transactions of the Royal Society of London |volume=53 |year=1763 |pages=370–418 |title=An Essay towards solving a Problem in the Doctrine of Chance. By the late Rev. Mr. Bayes, communicated by Mr. Price, in a letter to John Canton, A.M.F.R.S.|author1=Bayes, Thomas |author2=Price, Richard |name-list-style=amp |doi-access=free }}</ref> and contains Bayes' theorem. Price wrote an introduction to the paper that provides some of the philosophical basis of [[Bayesian statistics]] and chose one of the two solutions Bayes offered. In 1765, Price was elected a Fellow of the Royal Society in recognition of his work on Bayes's legacy.<ref name="Holland46">Holland, pp.&nbsp;46–7.</ref><ref>{{cite book |first = Richard |last = Price |title=Price: Political Writings |url = https://books.google.com/books?id=xdH-gjy2vzUC&pg=PR23 |access-date=16 June 2013 |year=1991 |publisher = Cambridge University Press |isbn = 978-0521409698 |page = xxiii }}</ref> On 27 April, a letter sent to his friend [[Benjamin Franklin]] was read out at the Royal Society, and later published, in which Price applies this work to population and computing 'life-annuities'.<ref name="EB1911">{{harvnb|Mitchell|1911|p=314}}.</ref>

Independently of Bayes, [[Pierre-Simon Laplace]] used conditional probability to formulate the relation of an updated [[posterior probability]] from a prior probability, given evidence. He reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work, and summarized his results in ''[[Théorie analytique des probabilités]]'' (1812).{{NoteTag |1 = Laplace refined Bayes's theorem over a period of decades:
* Laplace announced his independent discovery of Bayes' theorem in: Laplace (1774) "Mémoire sur la probabilité des causes par les événements", "Mémoires de l'Académie royale des Sciences de MI (Savants étrangers)", '''4''': 621–656. Reprinted in: Laplace, "Oeuvres complètes" (Paris, France: Gauthier-Villars et fils, 1841), vol. 8, pp.&nbsp;27–65. Available on-line at: [http://gallica.bnf.fr/ark:/12148/bpt6k77596b/f32.image Gallica]. Bayes' theorem appears on p.&nbsp;29.
* Laplace presented a refinement of Bayes' theorem in: Laplace (read: 1783 / published: 1785) "Mémoire sur les approximations des formules qui sont fonctions de très grands nombres", "Mémoires de l'Académie royale des Sciences de Paris", 423–467. Reprinted in: Laplace, "Oeuvres complètes" (Paris, France: Gauthier-Villars et fils, 1844), vol. 10, pp.&nbsp;295–338. Available on-line at: [http://gallica.bnf.fr/ark:/12148/bpt6k775981/f218.image.langEN Gallica]. Bayes' theorem is stated on page 301.
* See also: Laplace, "Essai philosophique sur les probabilités" (Paris, France: Mme. Ve. Courcier [Madame veuve (i.e., widow) Courcier], 1814), [https://books.google.com/books?id=rDUJAAAAIAAJ&pg=PA10 page 10]. English translation: Pierre Simon, Marquis de Laplace with F. W. Truscott and F. L. Emory, trans., "A Philosophical Essay on Probabilities" (New York, New York: John Wiley & Sons, 1902), [https://google.com/books?id=WxoPAAAAIAAJ&pg=PA15#v=onepage p. 15].}}<ref>{{cite book |title = Classical Probability in the Enlightenment |first=Lorraine |last=Daston |publisher=Princeton Univ Press |year=1988 |page=268 |isbn=0691084971 |url = https://books.google.com/books?id=oq8XNbKyUewC&pg=PA268 }}</ref> The [[Bayesian probability|Bayesian interpretation]] of probability was developed mainly by Laplace.<ref>{{cite book |last1=Stigler |first1=Stephen M. |chapter=Inverse Probability |pages=99–138 |chapter-url={{Google books|M7yvkERHIIMC|page=99|plainurl=yes}} |title=The History of Statistics: The Measurement of Uncertainty Before 1900 |date=1986 |publisher=Harvard University Press |isbn=978-0674403413 }}</ref>

About 200 years later, [[Harold Jeffreys|Sir Harold Jeffreys]] put Bayes's algorithm and Laplace's formulation on an [[axiomatic system|axiomatic]] basis, writing in a 1973 book that Bayes' theorem "is to the theory of probability what the [[Pythagorean theorem]] is to geometry".<ref name="Jeffreys1973">{{cite book |last=Jeffreys |first=Harold |author-link=Harold Jeffreys |year=1973 |title=Scientific Inference |url=https://archive.org/details/scientificinfere0000jeff |url-access=registration |publisher=[[Cambridge University Press]] |edition=3rd |isbn=978-0521180788 |page=[https://archive.org/details/scientificinfere0000jeff/page/31 31]}}</ref>

[[Stephen Stigler]] used a Bayesian argument to conclude that Bayes' theorem was discovered by [[Nicholas Saunderson]], a blind English mathematician, some time before Bayes,<ref>{{cite journal |last = Stigler |first = Stephen M. |year = 1983 |title = Who Discovered Bayes' Theorem? |journal = The American Statistician |volume = 37 |issue = 4 |pages = 290–296 |doi = 10.1080/00031305.1983.10483122 }}</ref><ref name="Stats, Data and Models">{{cite book |title = Stats, Data and Models |last1 = de Vaux |first1=Richard |last2=Velleman |first2=Paul |last3=Bock |first3=David |year=2016 |publisher=Pearson |isbn = 978-0321986498 |edition=4th |pages=380–381 }}</ref> but that is disputed.<ref>{{cite journal |last = Edwards |first = A. W. F. | year = 1986 | title = Is the Reference in Hartley (1749) to Bayesian Inference? |journal = The American Statistician |volume = 40 |issue = 2 |pages = 109–110 |doi = 10.1080/00031305.1986.10475370 }}</ref> 

Martyn Hooper<ref>{{cite journal |last = Hooper |first = Martyn |s2cid = 153704746 | year = 2013 |title = Richard Price, Bayes' theorem, and God |journal = Significance |volume = 10 |issue = 1 |pages = 36–39 |doi = 10.1111/j.1740-9713.2013.00638.x |doi-access = free }}</ref> and Sharon McGrayne<ref name="mcgrayne2011theory">{{cite book |last = McGrayne |first = S. B. |title = The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines & Emerged Triumphant from Two Centuries of Controversy |url = https://archive.org/details/theorythatwouldn0000mcgr |url-access = registration |publisher=[[Yale University Press]] |year=2011 |isbn=978-0300188226 }}</ref> have argued that Richard Price's contribution was substantial:
{{Blockquote|By modern standards, we should refer to the Bayes–Price rule. Price discovered Bayes's work, recognized its importance, corrected it, contributed to the article, and found a use for it. The modern convention of employing Bayes's name alone is unfair but so entrenched that anything else makes little sense.<ref name="mcgrayne2011theory" />|sign=|source=}}

[[F. Thomas Bruss]] reviewed Bayes' work "An essay towards solving a problem in the doctrine of chances" as communicated by Price.<ref>{{cite journal|last = Bruss |first = F. Thomas|year = 2014 |title = 250 years of "An essay towards solving a problem in the doctrine of chances" communicated by Price to the Royal Society. |journal = Jahresbericht der Deutschen Mathematiker-Vereinigung |volume = 115|issue =3 |pages = 129–133| doi=10.1365/s13291-013-0069-z }}</ref> He agrees with Stigler's fine analysis in many points, but not as far as the question of priority is concerned. Bruss underlines the intuitive part of Bayes' formula and adds independent arguments of Bayes' probable motivation for his work. He concludes that, unless the contrary is really proven, we are entitled to be faithful to the name "Bayes' Theorem" or "Bayes' formula".