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Thomas Bayes ( ; 1701 7 April 1761) was an English
statistician A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, and statisticians may wor ...
, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name:
Bayes' theorem In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exa ...
. Bayes never published what would become his most famous accomplishment; his notes were edited and published posthumously by Richard Price.


Biography

Thomas Bayes was the son of London Presbyterian minister Joshua Bayes, and was possibly born in Hertfordshire. He came from a prominent nonconformist family from
Sheffield Sheffield is a city status in the United Kingdom, city in South Yorkshire, England, whose name derives from the River Sheaf which runs through it. The city serves as the administrative centre of the City of Sheffield. It is Historic counties o ...
. In 1719, he enrolled at the
University of Edinburgh The University of Edinburgh ( sco, University o Edinburgh, gd, Oilthigh Dhùn Èideann; abbreviated as ''Edin.'' in post-nominals) is a public research university based in Edinburgh, Scotland. Granted a royal charter by King James VI in 15 ...
to study
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
and theology. On his return around 1722, he assisted his father at the latter's chapel in London before moving to Tunbridge Wells, Kent, around 1734. There he was minister of the Mount Sion Chapel, until 1752. He is known to have published two works in his lifetime, one theological and one mathematical: #''Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures'' (1731) #''An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of The Analyst'' (published anonymously in 1736), in which he defended the logical foundation of
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...
's
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
("fluxions") against the criticism by
George Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley ( Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immate ...
, a bishop and noted philosopher, the author of '' The Analyst'' Bayes was elected as a
Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural knowledge, including mathematic ...
in 1742. His nomination letter was signed by Philip Stanhope, Martin Folkes,
James Burrow Sir James Burrow (28 November 1701 – 5 November 1782 at Starborough Castle, Lingfield, Surrey), was a legal reporter at Inner Temple, London, and was vice president and twice briefly president of the Royal Society. He was knighted in 1773. ...
,
Cromwell Mortimer Cromwell Mortimer FRS (June 1702 – 7 January 1752) was a British physician, antiquary and second secretary of the Royal Society from 1730 to 1752. Early life Mortimer was the second son of John Mortimer of Topping Hall in Hatfield Peverel, Es ...
, and John Eames. It is speculated that he was accepted by the society on the strength of the ''Introduction to the Doctrine of Fluxions'', as he is not known to have published any other mathematical work during his lifetime. In his later years he took a deep interest in probability. Historian Stephen Stigler thinks that Bayes became interested in the subject while reviewing a work written in 1755 by Thomas Simpson, but
George Alfred Barnard George Alfred Barnard (23 September 1915 – 9 August 2002) was a British statistician known particularly for his work on the foundations of statistics and on quality control. Biography George Barnard was born in Walthamstow, Lond ...
thinks he learned mathematics and probability from a book by
Abraham de Moivre Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He mov ...
. Others speculate he was motivated to rebut
David Hume David Hume (; born David Home; 7 May 1711 NS (26 April 1711 OS) – 25 August 1776) Cranston, Maurice, and Thomas Edmund Jessop. 2020 999br>David Hume" ''Encyclopædia Britannica''. Retrieved 18 May 2020. was a Scottish Enlightenment phil ...
's argument against believing in miracles on the evidence of testimony in '' An Enquiry Concerning Human Understanding''. His work and findings on probability theory were passed in manuscript form to his friend Richard Price after his death. By 1755 he was ill and by 1761 had died in Tunbridge Wells. He was buried in Bunhill Fields burial ground in Moorgate, London, where many nonconformists lie. In 2018, the
University of Edinburgh The University of Edinburgh ( sco, University o Edinburgh, gd, Oilthigh Dhùn Èideann; abbreviated as ''Edin.'' in post-nominals) is a public research university based in Edinburgh, Scotland. Granted a royal charter by King James VI in 15 ...
opened a £45 million research centre connected to its informatics department named after its alumnus, Bayes. In April 2021, it was announced that
Cass Business School Bayes Business School, formerly known as Cass Business School, is the business school of City, University of London, located in St Luke's, just to the north of the City of London. It was established in 1966, and it is consistently ranked as on ...
, whose
City of London The City of London is a city, ceremonial county and local government district that contains the historic centre and constitutes, alongside Canary Wharf, the primary central business district (CBD) of London. It constituted most of London f ...
campus is on
Bunhill Row Bunhill Row is a street located in St Luke's, London Borough of Islington, London. The street runs north–south from Old Street to Chiswell Street. On the east side are the cemetery of Bunhill Fields and the open space of the Honourable Art ...
, was to be renamed after Bayes.


Bayes' theorem

Bayes's solution to a problem of inverse probability was presented in ''
An Essay towards solving a Problem in the Doctrine of Chances ''An Essay towards solving a Problem in the Doctrine of Chances'' is a work on the mathematical theory of probability by Thomas Bayes, published in 1763, two years after its author's death, and containing multiple amendments and additions due to ...
'', which was read to the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, r ...
in 1763 after Bayes's death. Richard Price shepherded the work through this presentation and its publication in the ''Philosophical Transactions of the Royal Society of London'' the following year. This was an argument for using a uniform prior distribution for a binomial parameter and not merely a general postulate. This essay gives the following theorem (stated here in present-day terminology).
Suppose a quantity ''R'' is uniformly distributed between 0 and 1. Suppose each of ''X''1, ..., ''X''''n'' is equal to either 1 or 0 and the
conditional probability In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...
that any of them is equal to 1, given the value of ''R'', is ''R''. Suppose they are conditionally independent given the value of ''R''. Then the conditional probability distribution of ''R'', given the values of ''X''1, ..., ''X''''n'', is : \frac r^S (1-r)^ \, dr \quad \text0\le r\le 1, \text S=X_1+\cdots+X_n.
Thus, for example, : \Pr(R \le r_0 \mid X_1,\ldots,X_n) = \frac \int_0^ r^S (1-r)^ \, dr. This is a special case of
Bayes' theorem In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exa ...
. In the first decades of the eighteenth century, many problems concerning the probability of certain events, given specified conditions, were solved. For example: given a specified number of white and black balls in an urn, what is the probability of drawing a black ball? Or the converse: given that one or more balls has been drawn, what can be said about the number of white and black balls in the urn? These are sometimes called " inverse probability" problems. Bayes's ''Essay'' contains his solution to a similar problem posed by
Abraham de Moivre Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He mov ...
, author of ''
The Doctrine of Chances ''The Doctrine of Chances'' was the first textbook on probability theory, written by 18th-century French mathematician Abraham de Moivre and first published in 1718.. De Moivre wrote in English because he resided in England at the time, havi ...
'' (1718). In addition, a paper by Bayes on
asymptotic series In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...
was published posthumously.


Bayesianism

Bayesian probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification o ...
is the name given to several related interpretations of
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
as an amount of epistemic confidence – the strength of beliefs, hypotheses etc. – rather than a frequency. This allows the application of probability to all sorts of propositions rather than just ones that come with a reference class. "Bayesian" has been used in this sense since about 1950. Since its rebirth in the 1950s, advancements in computing technology have allowed scientists from many disciplines to pair traditional Bayesian statistics with
random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb ...
techniques. The use of the
Bayes theorem In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For ex ...
has been extended in science and in other fields. Bayes himself might not have embraced the broad interpretation now called Bayesian, which was in fact pioneered and popularised by
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
;Stigler, Stephen M. (1986) ''The history of statistics.'', Harvard University press. pp 97–98, 131. it is difficult to assess Bayes's philosophical views on probability, since his essay does not go into questions of interpretation. There, Bayes defines ''probability'' of an event as "the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening" (Definition 5). In modern
utility As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
theory, the same definition would result by rearranging the definition of expected utility (the probability of an event times the payoff received in case of that event – including the special cases of buying risk for small amounts or buying security for big amounts) to solve for the probability. As Stigler points out, this is a subjective definition, and does not require repeated events; however, it does require that the event in question be observable, for otherwise it could never be said to have "happened". Stigler argues that Bayes intended his results in a more limited way than modern Bayesians. Given Bayes's definition of probability, his result concerning the parameter of a
binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no qu ...
makes sense only to the extent that one can bet on its observable consequences. The philosophy of Bayesian statistics is at the core of almost every modern estimation approach that includes conditioned probabilities, such as sequential estimation, probabilistic machine learning techniques, risk assessment, simultaneous localization and mapping, regularization or information theory. The rigorous axiomatic framework for probability theory as a whole, however, was developed 200 years later during the early and middle 20th century, starting with insightful results in ergodic theory by Plancherel in 1913.


See also

* Alison Gopnik * Bayesian epistemology * Bayesian inference * Bayesian network *
Bayesian statistics Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about the event, ...
* Development of doctrine * ''
Grammar of Assent ''An Essay in Aid of a Grammar of Assent'' (commonly abbreviated to the last three words) is John Henry Newman's seminal book on the philosophy of faith."NEWMAN, John Henry", in ''Chambers Biographical Dictionary'' (1990), Edinburgh: Chambers. ...
'' * Judea Pearl *
List of things named after Thomas Bayes Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...
* Probabiliorism * Theory-theory


Notes


References


Citations


Sources

* Thomas Bayes,
An essay towards solving a Problem in the Doctrine of Chances.
Bayes's essay in the original notation. * Thomas Bayes, 1763,
An essay towards solving a Problem in the Doctrine of Chances.
Bayes's essay as published in the Philosophical Transactions of the Royal Society of London, Vol. 53, p. 370, on Google Books. * Thomas Bayes, 1763,
A letter to John Canton
" ''Phil. Trans. Royal Society London'' 53: 269–71. * D. R. Bellhouse,. * D. R. Bellhouse, 2004,
The Reverend Thomas Bayes, FRS: A Biography to Celebrate the Tercentenary of His Birth
" ''Statistical Science'' 19 (1): 3–43. *
F. Thomas Bruss Franz Thomas Bruss is Emeritus Professor of Mathematics at the Université Libre de Bruxelles, where he had been director of "Mathématiques Générales" and co-director of the probability chair, and where he continues his research as invited profe ...
(2013), "250 years of '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.' ", , Jahresbericht der Deutschen Mathematiker-Vereinigung, Springer Verlag, Vol. 115, Issue 3–4 (2013), 129–133. * Dale, Andrew I. (2003.) "Most Honourable Remembrance: The Life and Work of Thomas Bayes". . Springer, 2003. * ____________. "An essay towards solving a problem in the doctrine of chances" in Grattan-Guinness, I., ed., ''Landmark Writings in Western Mathematics''. Elsevier: 199–207. (2005). * Michael Kanellos
"18th-century theory is new force in computing"
CNET News, 18 February 2003. * McGrayne, Sharon Bertsch. (2011). ''The Theory That Would Not Die: How Bayes's Rule Cracked The Enigma Code, Hunted Down Russian Submarines, & Emerged Triumphant from Two Centuries of Controversy.'' New Haven: Yale University Press.
OCLC 670481486
* Stigler, Stephen M.br>"Thomas Bayes's Bayesian Inference,"
''
Journal of the Royal Statistical Society The ''Journal of the Royal Statistical Society'' is a peer-reviewed scientific journal of statistics. It comprises three series and is published by Wiley for the Royal Statistical Society. History The Statistical Society of London was founde ...
'', Series A, 145:250–258, 1982. * ____________. "Who Discovered Bayes's Theorem?" ''The American Statistician'', 37(4):290–296, 1983.


External links


The will of Thomas Bayes 1761

Author profile
in the database zbMATH * Full text o
Divine Benevolence: Or, An Attempt to Prove that the Principal End of the Divine Providence and Government is the Happiness of His Creatures...
* Full text o
An Introduction to the Doctrine of Fluxions, And Defence of the Mathematicians Against the Objections of the Author of the Analyst, So Far as They are Designed to Affect Their General Methods of Reasoning
{{DEFAULTSORT:Bayes, Thomas 1701 births 1761 deaths 18th-century English mathematicians 18th-century English non-fiction writers 18th-century essayists 18th-century philosophers 18th-century Presbyterian ministers Alumni of the University of Edinburgh Bayesian statisticians Burials at Bunhill Fields English Christians English male non-fiction writers English philosophers English Presbyterian ministers English statisticians Epistemologists Fellows of the Royal Society Mathematicians from London People from Royal Tunbridge Wells Philosophers of mathematics Philosophers of religion Probability theorists Rationality theorists