Thomas Bayes ( BAYZ; c. 1701 – 7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem.
Bayes never published what would become his most famous accomplishment; his notes were edited and published posthumously by Richard Price.
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. 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 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's calculus ("fluxions") against the criticism by George Berkeley, a bishop and noted philosopher, the author of The Analyst
Bayes was elected as a Fellow of the Royal Society in 1742. His nomination letter was signed by Philip Stanhope, Martin Folkes, James Burrow, Cromwell Mortimer, 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 thinks he learned mathematics and probability from a book by Abraham de Moivre. Others speculate he was motivated to rebut David Hume'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, he 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 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, whose City of London campus is on Bunhill Row, was to be renamed after Bayes.
Bayes's solution to a problem of inverse probability was presented in An Essay Towards Solving a Problem in the Doctrine of Chances, 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 gives the following theorem (stated here in present-day terminology).
Suppose a quantity R is uniformly distributed between 0 and 1. Suppose each of X1, ..., Xn is equal to either 1 or 0 and the conditional probability 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 X1, ..., Xn, is
{\displaystyle {\frac {(n+1)!}{S!(n-S)!}}r^{S}(1-r)^{n-S}\,dr\quad {\text{for }}0\leq r\leq 1,{\text{ where }}S=X_{1}+\cdots +X_{n}.}
{\displaystyle \Pr(R\leq r_{0}\mid X_{1},\ldots ,X_{n})={\frac {(n+1)!}{S!(n-S)!}}\int _{0}^{r_{0}}r^{S}(1-r)^{n-S}\,dr.}
This is a special case of the 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? 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, author of The Doctrine of Chances (1718).