# Bayes' theorem

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In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule) is a result that is of importance in the mathematical manipulation of conditional probabilities. It is a result that derives from the more basic axioms of probability.

When applied, the probabilities involved in Bayes' theorem may have any of a number of probability interpretations. In one of these interpretations, the theorem is used directly as part of a particular approach to statistical inference. In particular, with the Bayesian interpretation of probability, the theorem expresses how a subjective degree of belief should rationally change to account for evidence: this is Bayesian inference, which is fundamental to Bayesian statistics. However, Bayes' theorem has applications in a wide range of calculations involving probabilities, not just in Bayesian inference.

Bayes' theorem is named after Thomas Bayes (Template:IPAc-en; 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.

Sir Harold Jeffreys wrote that Bayes' theorem “is to the theory of probability what Pythagoras's theorem is to geometry”.