Everything about Thomas Bayes totally explained
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 isn't 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 while reading 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.
It isn't 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 doesn't 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 it's happening
In modern
utility theory we'd 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 doesn't 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.
Further Information
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