Thomas Bayes (1701-1761)

Reverend Thomas Bayes was a nonconformist minister, who studied at Edinburgh University. He was interested in mathematics and corresponded with leading mathematicians of his time. He only published two works during his life, both unrelated to probability: one in 1731 called Divine Benevolence: or, An Attempt to prove that the Principal End of the Divine Providence and Government is the Happiness of his Creatures and one in 1736 called An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians against the objections of the Author of the Analyst. In the latter, he refuted Bishop George Berkeley's The Analyst, which used surface level understanding of calculus ideas to argue that scientific development threatened religion. Bayes built his argument on the foundational ideas of calculus. Both of Bayes' works were focused on religion, but when he died, another minister, at his family's request, looked at his unpublished papers related to mathematics and found An Essay towards Solving a Problem in the Doctrine of Chances and published it. At first, it received very little notice, but then gathered both positive and negative recognition because of its controversial central idea called inverse probability. The idea of inverse probability is manifested today in both Bayesian statistics and the more common Bayes' Rule, and it is possible that Bayes never published this work because of the controversy he believed it would bring.

Bayes' Rule has since been used in several applications, including determining the causal relationship between smoking and lung cancer. Another application was determining who wrote some of the Federalist papers since a few were attributed to both James Madison and Alexander Hamilton. To determine who actually wrote them, statisticians analyzed the prevalence of non-content words such as if, when, and over in the papers in question and the possible authors' other writing. They found, for example, that Madison used the word upon $0.23$ times in $1,000$ words and Hamilton used upon an average of $3.24$ times in $1,000$ words. These estimates come from a distribution for each potential author of the averages in all their works. Each distribution can be used to calculate the probability that the potential author wrote the given paper.

Bayes also contributed to the development of personal probability, which is the idea that there is no such thing as a proven scientific fact. Instead, what are commonly thought of as facts are statements that scientists have identified as highly probable. The Bayesian view of statistics has led to different approaches, including the Savage-de Finetti which asserts that each individual has their own set of personal probabilities. In another, the Keynes view, probability is the overall confidence an educated person in a given culture has in a particular statement. The most famous idea credited to Bayes is based on the assumption that there is a starting personal probability that is modified by data.