Thomas Bayes, a Presbyterian minister, expressed a method of inductive inference in a precise and quantitative form, which lead to the development of Bayesian statistics, or Bayesian inference. His stature as a mathematician is based on only two short mathematical papers, both of which were published posthumously by the Royal Society of London. The first paper demonstrates what may be the first recognition of asymptotic behavior by series expansions. The second and far more important paper addresses a problem with continuing application in most areas of human endeavor. In this paper, Bayes discusses the estimation of future occurrences of an event, given knowledge of the history of the event--that it has occurred a number of times and failed a number of times. This work continues to spawn mathematical research, and provides the foundations for Bayesian statistical estimation, used today on such diverse problems as electoral polling or estimating time to failure of mechanical devices.
Little is known of Bayes' childhood. Some sources note that he was privately educated, while others state that he received a liberal education in preparation for the ministry. Bayes was the eldest of six children of Joshua and Ann Carpenter Bayes. His father was a Nonconformist minister, one of the first seven publicly ordained in England. Thomas' paternal grandfather, Joshua Bayes, had been a cutler and town collector in Sheffield. Thomas' place of birth is usually listed as London, but one biographer suggests that he was born in Hertfordshire, where his peripatetic father supposedly preached at the time of his birth. Unfortunately, the appropriate parish records of 1700-1706 have been lost. Thomas' epitaph in the family vault at Moorgate states his age at death as 59 in April 1761, placing his birth in 1701 or 1702.
Educated for the Ministry
Andrew I. Dale argues persuasively that Bayes was educated at Edinburgh University. Thomas Bayes' name appears in a 1719 catalogue of manuscripts in the Edinburgh University Library, and in a number of other records at the University over the period 1720-1722, including class lists and a list of theologues. The Bayes signature at Edinburgh matches closely that of the Royal Society records. Bayes received only licensure for the ministry at Edinburgh, but he was ordained during or before 1727, and is included in Dr. John Evans' 1727 list of "Approved Ministers of the Presbyterian Denomination." Bayes assisted his father at his ministry in Leather Lane for some years from 1728 before succeeding the Rev. John Archer as minister at Tunbridge Wells, Kent. He spent the remainder of his life in Tunbridge Wells as Presbyterian minister of the Mount Sion meeting house.
Defense of Newton Is Catalyst for Election to Royal Society
Two years after the publication of The Analyst; or, a Discourse addressed to an Infidel Mathematician (1734), George Berkeley's famous attack of Isaac Newton's work on fluxions (differentials), an anonymous tract was published that answered Berkeley and vigorously defended Newton's work. The tract, titled An Introduction to the Doctrine of Fluxions and Defence of the Mathematicians against the Objections of the Author of the Analyst, was widely attributed to Bayes and was probably the reason behind his election as a Fellow of the Royal Society in 1742. In it, he addresses the "business of the mathematician," and stated that "[he] is not inquiring how things are in matter of fact, but supposing things to be in a certain way, what are the consequences to be deduced from them; and all that is to be demanded of him is, that his suppositions be intelligible, and his inferences just from the suppositions he makes." The proposal for Bayes' election to the Royal Society read in part that he was "well skilled in geometry and all parts of mathematical and philosophical learning." It was signed by Eames, James Burrow, Cromwell Mortimer, Martin Folkes, and Earl Stanhope.
Work Published Posthumously
Bayes retired from the ministry around 1750. He died at Tunbridge Wells on April 17, 1761, leaving a fairly substantial estate, and was buried in the family vault at Bunhill Fields Burial Ground at Moorgate. Upon his death, Bayes' family asked his friend, the Unitarian Reverend Richard Price, to examine his papers. Among them Price found Bayes' work on probability.
In 1764, Bayes' paper on the Stirling-De Moivre Theorem, dealing with series expansions, was published in the Philosophical Transactions of the Royal Society. Price declared the problem in inductive reasoning stated by Bayes to be central "to the argument taken from final causes for the existence of the Deity." The same issue of the Philosophical Transactions contains a second piece by Bayes, "An Essay towards Solving a Problem in the Doctrine of Chances," presented also by Price with his preface, footnotes and appendix. The problem posed in the essay, wrote Price, is "to find out a method by which we might judge concerning the probabilitythat an event has to happen, in given circumstances, upon supposition that we know nothing concerning it but that, under the same circumstances, it has happened a certain number of times, and failed a certain other number of times." The essay was followed in the next volume by "A Demonstration of the Second Rule in the Essay. . .," a continuation of Bayes' results which were further developed by Price.
A notebook belonging to Bayes has been preserved in the London records room of the Equitable Life Assurance Society, due to the action of Price and his nephew William Morgan, an actuary. Among other curiosities, the notebook includes the key to a system of shorthand, details of an electrifying machine, lists of English weights and measures, notes on topics in mathematics, natural philosophy and celestial mechanics, and a proof of one of the rules in the "Essay" that was published after Bayes' death.
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