Thomas Bayes

# Overview
Thomas Bayes (c. 1702  April 17,1761) was a British mathematician and Presbyterian minister, known for having formulated a special case of Bayes' theorem, which was published posthumously.

# Biography
Thomas Bayes was born in London.  In 1719 he enrolled at the University of Edinburgh to study logic and theology: as a Nonconformist, Oxford and Cambridge were closed to him.
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.
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. He was intrerested in probability after reading a paper by Simpson.
It is not at all clear that Bayes himself would have embraced the very broad interpretation now called Bayesian.
It is difficult to assess Bayes' philosophical views on probability,
as the only direct evidence is his essay, which does not go into questions of interpretation. In the essay, Bayes defines probability as follows (Definition 5).
In modern utility theory we would say that expected utility is - sometimes, because buying risk for small amounts or buying security for big amounts also happens - the probability of an event times the payoff received in case of that event.
Rearranging that to solve for the probability, we obtain Bayes' definition.
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".  (Some would argue, however, that things can happen without being observable.)
Thus it can be argued, as Stigler does,
that Bayes intended his results in a rather 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.