The Important Theorem of Thomas Bayes

Thomas Bayes (1701 - 1761) was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem

Thomas Bayes (1701 – 1761)

On April 17, 1761, English mathematician and Presbyterian minister Thomas Bayes passed away. He is best known as name giver of the Bayes’ theorem, of which he had developed a special case. It expresses (in the Bayesian interpretation) how a subjective degree of belief should rationally change to account for evidence, and finds application in in fields including science, engineering, economics (particularly microeconomics), game theory, medicine and law.

Thomas Bayes was born into a prominent family from Sheffield in 1701 and enrolled at the University of Edinburgh in 1719. He began studying logic and theology, assisting his father at the non-conformist chapel in London. He was ordained in 1727 and moved to Box Lane Chapel, Bovington, about 25 miles from London. In later years, Bayes became minister of the Mount Sion chapel.

We do know that in 1719 Bayes matriculated at the University of Edinburgh where he studied logic and theology. He had to choose a Scottish university if he was to obtain his education without going overseas since, at this time, Nonconformists were not allowed to matriculate at Oxford or Cambridge.[5] Even though Bayes is known to have published only two major works, they were quite influential. The first, published in 1731, was titled ‘Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures‘ was rather dedicated to the field of theology. Five years later however, Bayes published anonymously ‘An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst‘. This work was rather mathematical and in it, Bayes defended Isaac Newton‘s theories on calculus and its logical foundations.[6]

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. In following years, Bayes’ interest in probability theories grew and his interesting ideas and findings were collected in his manuscripts and most of them only became known after his passing.

Unfortunately, Bayes’ most important and most influential work was published after his death. The ‘Essay towards solving a Problem in the Doctrine of Chances‘ was read to the Royal Society in 1763. The work contained a statement of a special case of probability now called Bayes’ theorem. The theorem can be seen as a “way of understanding how the probability that a theory is true is affected by a new piece of evidence”. Through the years, it has been helpful in a variety of scientific fields, and is often used to clarify the relationship between theory and evidence. Richard Price, who once published Bayes’ papers after his death believed that Bayes’ theorem could prove the existence of God. This technically combines Bayes’ research fields in mathematics and theology and Price’s claim opened up a wide field of discussion for future scientists and it still depicts a debate topic up to this day.

In probability theory and statistics, Bayes’ theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person’s age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person’s age. One of the many applications of Bayes’ theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in Bayes’ theorem may have different probability interpretations. With the Bayesian probability interpretation the theorem expresses how a subjective degree of belief should rationally change to account for availability of related evidence. Bayesian inference is fundamental to Bayesian statistics. After Bayes initial formulation, the theory was further developed by Pierre-Simon Laplace,[7] who first published the modern formulation in his 1812 “Théorie analytique des probabilités”. Sir Harold Jeffreys put Bayes’ algorithm and Laplace’s formulation on an axiomatic basis. Jeffreys wrote that Bayes’ theorem “is to the theory of probability what the Pythagorean theorem is to geometry

At yovisto academic video search, you may enjoy a video lecture on Basic Probability Theory by Professor Dr Faber at Zurich.

References and Further Reading:

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