Thomas Bayes
Encyclopedia
Thomas Bayes (c. 1701 7 April 1761) was an English
mathematician
and Presbyterian minister, known for having formulated a specific case of the theorem that bears his name: Bayes' theorem
. Bayes never published what would eventually become his most famous accomplishment; his notes were edited and published after his death by Richard Price
.
. In 1719, he enrolled at the University of Edinburgh
to study logic
and theology
. On his return around 1722, he assisted his father at the latter's non-conformist chapel in London before moving to Tunbridge Wells
, Kent
around 1734. There he became minister of the Mount Sion chapel, until 1752.
He is known to have published two works in his lifetime, one theological and one mathematical:
It is speculated that Bayes was elected as a Fellow of the Royal Society in 1742 on the strength of the Introduction to the Doctrine of Fluxions, as he is not known to have published any other mathematical works during his lifetime.
In his later years he took a deep interest in probability. Stephen Stigler
feels that he became interested in the subject while reviewing a work written in 1755 by Thomas Simpson
, but George Alfred Barnard
thinks he learned mathematics and probability from a book by de Moivre
. 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 in 1761 had died in Tunbridge. He was buried in Bunhill Fields
Cemetery in Moorgate, London where many Nonconformists lie.
which was read to the Royal Society 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 contains a statement of a special case of Bayes' theorem
.
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?
Attention soon turned to the converse of such a problem:
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.
The Essay of Bayes contains his solution to a similar problem, posed by Abraham de Moivre
, author of The Doctrine of Chances
(1718).
In addition to the Essay Towards Solving a Problem,
a paper on asymptotic series was published posthumously.
is the name given to several related interpretations of probability
, which have in common the notion of probability as something like a partial belief, 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 1950s, advancements in computing technology have allowed scientists from many disciplines to pair traditional Bayesian statistics with random walk
techniques. The use of the Bayes theorem has been extended in science and in other fields.
Bayes himself might not have embraced the broad interpretation now called Bayesian. It is difficult to assess Bayes' philosophical views on probability, since his essay does not go into questions of interpretation. There Bayes defines probability as follows (Definition 5).
In modern utility theory, expected utility can (with qualifications, because buying risk for small amounts or buying security for big amounts also happen) be taken as the probability of an event times the payoff received in case of that event. Rearranging that to solve for the probability, Bayes' definition results. 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' definition of probability, his result concerning the parameter of a binomial distribution makes sense only to the extent that one can bet on its observable consequences.
External links
Kingdom of Great Britain
The former Kingdom of Great Britain, sometimes described as the 'United Kingdom of Great Britain', That the Two Kingdoms of Scotland and England, shall upon the 1st May next ensuing the date hereof, and forever after, be United into One Kingdom by the Name of GREAT BRITAIN. was a sovereign...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and Presbyterian minister, known for having formulated a specific case of the theorem that bears his name: Bayes' theorem
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
. Bayes never published what would eventually become his most famous accomplishment; his notes were edited and published after his death by Richard Price
Richard Price
Richard Price was a British moral philosopher and preacher in the tradition of English Dissenters, and a political pamphleteer, active in radical, republican, and liberal causes such as the American Revolution. He fostered connections between a large number of people, including writers of the...
.
Biography
Thomas Bayes was the son of London Presbyterian minister Joshua Bayes and perhaps born in HertfordshireHertfordshire
Hertfordshire is a ceremonial and non-metropolitan county in the East region of England. The county town is Hertford.The county is one of the Home Counties and lies inland, bordered by Greater London , Buckinghamshire , Bedfordshire , Cambridgeshire and...
. In 1719, he enrolled at the University of Edinburgh
University of Edinburgh
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
to study logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...
. On his return around 1722, he assisted his father at the latter's non-conformist chapel in London before moving to Tunbridge Wells
Royal Tunbridge Wells
Royal Tunbridge Wells is a town in west Kent, England, about south-east of central London by road, by rail. The town is close to the border of the county of East Sussex...
, Kent
Kent
Kent is a county in southeast England, and is one of the home counties. It borders East Sussex, Surrey and Greater London and has a defined boundary with Essex in the middle of the Thames Estuary. The ceremonial county boundaries of Kent include the shire county of Kent and the unitary borough of...
around 1734. There he became 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 NewtonIsaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
's calculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
("fluxions") against the criticism of George BerkeleyGeorge BerkeleyGeorge Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...
, author of The AnalystThe AnalystThe Analyst, subtitled "A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith", is a book published by...
It is speculated that Bayes was elected as a Fellow of the Royal Society in 1742 on the strength of the Introduction to the Doctrine of Fluxions, as he is not known to have published any other mathematical works during his lifetime.
In his later years he took a deep interest in probability. Stephen Stigler
Stephen Stigler
Stephen Mack Stigler is Ernest DeWitt Burton Distinguished Service Professor at the Department of Statistics of the University of Chicago. His research has focused on statistical theory of robust estimators and the history of statistics...
feels that he became interested in the subject while reviewing a work written in 1755 by Thomas Simpson
Thomas Simpson
Thomas Simpson FRS was a British mathematician, inventor and eponym of Simpson's rule to approximate definite integrals...
, but George Alfred Barnard
George Alfred Barnard
George Alfred Barnard was a British statistician known particularly for his work on the foundations of statistics and on quality control.-Biography:...
thinks he learned mathematics and probability from a book by de Moivre
Abraham de Moivre
Abraham de Moivre was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling...
. His work and findings on probability theory were passed in manuscript form to his friend Richard Price
Richard Price
Richard Price was a British moral philosopher and preacher in the tradition of English Dissenters, and a political pamphleteer, active in radical, republican, and liberal causes such as the American Revolution. He fostered connections between a large number of people, including writers of the...
after his death.
By 1755 he was ill and in 1761 had died in Tunbridge. He was buried in Bunhill Fields
Bunhill Fields
Bunhill Fields is a cemetery in the London Borough of Islington, north of the City of London, and managed by the City of London Corporation. It is about 4 hectares in extent, although historically was much larger....
Cemetery in Moorgate, London where many Nonconformists lie.
Bayes' theorem
Bayes' solution to a problem of "inverse probability" was presented in the An Essay towards solving a Problem in the Doctrine of ChancesAn 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 the Reverend Thomas Bayes, published in 1763, two years after its author's death. It included a statement of a special case of what is now called Bayes' theorem. In 18th-century...
which was read to the Royal Society in 1763 after Bayes's death. Richard Price
Richard Price
Richard Price was a British moral philosopher and preacher in the tradition of English Dissenters, and a political pamphleteer, active in radical, republican, and liberal causes such as the American Revolution. He fostered connections between a large number of people, including writers of the...
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 contains a statement of a special case of Bayes' theorem
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
.
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?
Attention soon turned to the converse of such a problem:
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
Inverse probability
In probability theory, inverse probability is an obsolete term for the probability distribution of an unobserved variable.Today, the problem of determining an unobserved variable is called inferential statistics, the method of inverse probability is called Bayesian probability, the "distribution"...
" problems.
The Essay of Bayes contains his solution to a similar problem, posed by Abraham de Moivre
Abraham de Moivre
Abraham de Moivre was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling...
, author of The Doctrine of Chances
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, having fled France to escape the persecution of Huguenots...
(1718).
In addition to the Essay Towards Solving a Problem,
a paper on asymptotic series was published posthumously.
Bayes and Bayesianism
Bayesian probabilityBayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
is the name given to several related interpretations of probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, which have in common the notion of probability as something like a partial belief, 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 1950s, advancements in computing technology have allowed scientists from many disciplines to pair traditional Bayesian statistics with random walk
Markov chain Monte Carlo
Markov chain Monte Carlo methods are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample of the...
techniques. The use of the Bayes theorem has been extended in science and in other fields.
Bayes himself might not have embraced the broad interpretation now called Bayesian. It is difficult to assess Bayes' philosophical views on probability, since his essay does not go into questions of interpretation. There Bayes defines probability as follows (Definition 5).
- The probability of any event is 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
In modern utility theory, expected utility can (with qualifications, because buying risk for small amounts or buying security for big amounts also happen) be taken as the probability of an event times the payoff received in case of that event. Rearranging that to solve for the probability, Bayes' definition results. 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' definition of probability, his result concerning the parameter of a binomial distribution makes sense only to the extent that one can bet on its observable consequences.
See also
- Bayesian brainBayesian brainBayesian brain is a term that is used to refer to the ability of the nervous system to operate in situations of uncertainty in a fashion that is close to the optimal prescribed by Bayesian statistics. This term is used in behavioural sciences and neuroscience and studies associated with this term...
- Bayesian spam filteringBayesian spam filteringBayesian spam filtering is a statistical technique of e-mail filtering. It makes use of a naive Bayes classifier to identify spam e-mail.Bayesian classifiers work by correlating the use of tokens , with spam and non spam e-mails and then using Bayesian inference to calculate a probability that an...
- Empirical Bayes
- Royal SocietyRoyal SocietyThe Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
– Bayes was elected to membership in the Society in 1742; and his nomination letter has been posted with other membership records at the Royal Society web site here. Those signing that nomination letter were: Philip StanhopePhilip Stanhope, 2nd Earl StanhopePhilip Stanhope, 2nd Earl Stanhope FRS was a British peer.The son of the 1st Earl Stanhope and Lucy Pitt, he succeeded to his father's titles in 1721. He was a Fellow of the Royal Society....
; Martin FolkesMartin FolkesMartin Folkes FRS , English antiquary, was born in London.He was educated at Saumur University and Clare College, Cambridge, where he so distinguished himself in mathematics that when only twenty-three years of age he was chosen a fellow of the Royal Society...
; James BurrowJames BurrowSir James Burrow, FRS, FSA , 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 MortimerCromwell MortimerCromwell Mortimer FRS was a British physician, antiquary and second secretary of the Royal Society from 1730-1752.-Early life:...
; John EamesJohn Eames-Life:He was a native of London. He was admitted to Merchant Taylors' School on 10 March 1696–7, and was subsequently trained for the dissenting ministry. He preached only once and seems never to have been ordained....
.
External links