Bayes' Theorem

In probability theory and statistics, Bayes' theorem (alternatively Bayes' law) is a theorem with two distinct interpretations. In the Bayesian interpretation, it expresses how a subjective degree of belief should rationally change to account for evidence. In the frequentist interpretation, it relates inverse representations of the probabilities concerning two events. In the Bayesian interpretation, Bayes' theorem is fundamental to Bayesian statistics, and has applications in fields including science, engineering, economics (particularly microeconomics), game theory, medicine and law. The application of Bayes' theorem to update beliefs is called Bayesian inference.

Bayes' theorem is named for Thomas Bayes (/ˈbeɪz/; 1701–1761), who first suggested using the theorem to update beliefs. His work was significantly edited and updated by Richard Price before it was posthumously read at the Royal Society. The ideas gained limited exposure until they were independently rediscovered and further developed by Laplace, who first published the modern formulation in his 1812 Théorie analytique des probabilités. Until the second half of the 20th century, the Bayesian interpretation was largely rejected by the mathematics community as unscientific. However, it is now widely accepted. This may have been due to the development of computing, which enabled the successful application of Bayesianism to many complex problems.