by A. W. F. Edwards

© Oxford University Press 2004 All rights reserved

**Bayes, Thomas** (1701?-1761), mathematician and Presbyterian minister, was probably born in Hertfordshire, the eldest in the family of four sons and three daughters of the Revd Joshua Bayes and his wife, Ann Carpenter. He entered Edinburgh University in 1719 and studied for three years, though without taking a degree. He was licensed to preach while in Edinburgh, ordained by 1727, and began his ministry as an assistant to his father, at the time minister of the Presbyterian meeting-house at Leather Lane, Hatton Garden, London. In 1731 he was appointed to minister in Tunbridge Wells at the meeting-house in Little Mount Sion, and in that year a tract attributed to him--Divine *benevolence, or, An attempt to prove that the principal end of the divine providence and government is the happiness of his creatures--was* published by John Noon.

In 1736 Noon published another anonymous tract which has always been attributed to Bayes--An *introduction to the doctrine of fluxions, and a defence of the mathematicians against the objections of the author of The analyst.* The author of *The Analyst, or, A Discourse Addressed to an Infidel Mathematician* (1734), George Berkeley (1685-1753), had attacked the practitioners of the Newtonian calculus, and impugned their piety; among the replies in defence of the mathematicians, that of Bayes was considered the most acute. Perhaps on this account he was elected a fellow of the Royal Society in 1742.

Bayes's fame rests on his posthumous paper, 'An essay towards solving a problem in the doctrine of chances', which was published in 1764 in the *Philosophical Transactions of the Royal Society* for the year 1763, having been 'found among [his] papers' by his friend the Revd Richard Price, who wrote an introduction for it. It contains a solution to the problem (in Bayes's words): *'Given* the number of times in which an unknown event has happened and failed: *Required* the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named' *(Philosophical Transactions,* 53, 1764, 370-418).

The problem is fundamental to statistical inference. It owes its origin to the *Ars conjectandi* of James Bernoulli (1713), and was extensively treated by Abraham de Moivre. Possibly Bayes learned of it directly from de Moivre, although he may equally well have encountered it first in his writings, or in *Observations on Man, his Frame, his Duty, and his Expectations* (1749) by David Hartley.

Bayes's tentative solution to the problem involves assuming that the lack of knowledge of the probability about which an inference is to be made may be represented by a uniform probability distribution, for which he gives an ingenious argument which has often been overlooked. The resulting type of statistical methodology is referred to as Bayesian and involves assigning a probability distribution to every unknown. Commonly rejected for much of the twentieth century, the use of such arguments has become more popular recently, though they remain controversial. Bayes's paper also includes a result which is a special case of a more general theorem in conditional probability which, anachronistically, has therefore become known as Bayes's theorem. In addition it contains some interesting contributions to the evaluation of the beta probability integral. Bayes's only other known published work is a short letter to John Canton on Stirling's approximation which was also printed in the *Philosophical Transactions* for 1763.

In 1752 Bayes retired from his ministry, but continued to live in Tunbridge Wells until his death. He died on 7 April 1761, and was buried in the Bayes and Cotton family vault in Bunhill Fields, the nonconformist burial-ground at Moorgate, London. He was unmarried. His will, executed on 12 December 1760, shows him to have been a man of substance. Richard Price described Bayes as the most ingenious man he ever knew, and William Whiston wrote that he was a very good mathematician. He is also said to have been a good Greek scholar and a poor preacher, and it is clear that he was retiring and modest.

A. W. F. EDWARDS

**Sources **

A. I. Dale, *A history of inverse probability from Thomas Bayes to Karl Pearson* (1991)

G. A. Barnard, 'Thomas Bayes--a biographical note', *Biometrika,* 45 (1958), 293-5; repr. in E. S. Pearson and M. G. Kendall, eds., *Studies in the history of probability and statistics,* 1 (1970)

J. D. Holland, 'The Reverend Thomas Bayes, FRS, 1702-61', *Journal of the Royal Statistical Society: series* A, 125 (1962), 451-61

S. M. Stigler, *The history of statistics* (1986)

D. R. Bellhouse, 'Tidbits on Thomas', *Institute of Mathematical Statistics Bulletin,* 19 (1990), 478-9

A. Hald, 'Evaluations of the beta probability integral by Bayes and Price', *Archive for History of Exact Sciences,* 41 (1990-91), 139-56

private information, 1995

burial register, 15 April 1761, Bunhill Fields burial-ground, Moorgate, London

**Archives **

Equitable Life Assurance Society, notebook

**Wealth at death **

approx. £2000-£3000: will, PRO

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