Thomas Bayes

Thomas Bayes (pronounced: ) (c. 1702 – 17 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, which was published posthumously.

Biography
Thomas Bayes was the son of a London Presbyterian minister, Joshua Bayes, born perhaps in Hertfordshire. In 1719 he enrolled at the University of Edinburgh to study logic and theology.

He is known to have published two works in his lifetime: Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731), and 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 against the criticism of George Berkeley, author of The Analyst.

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.

Some feel that he became interested in probability while reviewing a work written in 1755 by Thomas Simpson, but others think he learned mathematics and probability from a book by de Moivre.

Bayes died in Tunbridge Wells, Kent. He is buried in Bunhill Fields Cemetery in London where many Nonconformists are buried.

Bayes' theorem
Bayes' solution to a problem of "inverse probability" was presented in the Essay Towards Solving a Problem in the Doctrine of Chances (1764), published posthumously by his friend Richard Price in the Philosophical Transactions of the Royal Society of London. 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? These are sometimes called "forward probability" problems. 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? 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.

Bayes and Bayesianism
Bayesian probability 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.

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 (citation below) 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.