, showing the simple statement of Bayes’ theorem
probability theory and
statistics, Bayes’ theorem (alternatively Bayes’ law or Bayes' rule) 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
Bayes’ theorem is named after Reverend
Thomas Bayes (/; 1701–1761), who first provided an equation that allows new evidence to update beliefs in his
An Essay towards solving a Problem in the Doctrine of Chances (1763). It was further developed by
Pierre-Simon Laplace, 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".