Thomas Bayes

Thomas Bayes
The correct identification of this portrait has been questioned [1]
Born	c. 1702 London, England
Died	17 April 1761 (aged 59) Tunbridge Wells, Kent, England
Nationality	British
Religious stance	Presbyterian
Signature

Thomas Bayes (pronounced: ˈbeɪz), (c. 1702 – 17 April 1761) was a British 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 born in London. 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,^[2] but others think he learned mathematics and probability while reading a book by de Moivre.^[3]

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.

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).

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 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.

See also

• Empirical Bayes
• Bayesian brain

Notes

References

Further reading

• Thomas Bayes, "An essay towards solving a Problem in the Doctrine of Chances." Bayes's essay in the original notation.
• Thomas Bayes, 1763, "A letter to John Canton," Phil. Trans. Royal Society London 53: 269-71.
• D. R. Bellhouse, "On Some Recently Discovered Manuscripts of Thomas Bayes."
• D. R. Bellhouse, 2004, "The Reverend Thomas Bayes, FRS: A Biography to Celebrate the Tercentenary of His Birth," Statistical Science 19 (1): 3–43.
• Dale, A.I., 2005, "An essay towards solving a problem in the doctrine of chances" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 199-207.

External links

• Who was The Rev. Thomas Bayes?
• Biographical sketch of Thomas Bayes
• An Intuitive Explanation of Bayesian Reasoning (includes biography)
• The will of Thomas Bayes 1761