Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes (/beɪz/)) gives a mathematical rule for inverting conditional probabilities, allowing the probability of a cause to be found given its effect. For example, with Bayes' theorem, the probability that a patient has a disease given that they tested positive for that disease can be found using the probability that the test yields a positive result when the disease is present. The theorem was developed in the 18th century by Bayes and independently by Pierre-Simon Laplace.

One of Bayes' theorem's many applications is Bayesian inference, an approach to statistical inference, where it is used to invert the probability of observations given a model configuration (i.e., the likelihood function) to obtain the probability of the model configuration given the observations (i.e., the posterior probability).

Bayes' theorem is named after Thomas Bayes, a minister, statistician, and philosopher. Bayes used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter. His work was published in 1763 as An Essay Towards Solving a Problem in the Doctrine of Chances. Bayes studied how to compute a distribution for the probability parameter of a binomial distribution (in modern terminology). After Bayes's death, his family gave his papers to a friend, the minister, philosopher, and mathematician Richard Price.

Price significantly edited the unpublished manuscript for two years before sending it to a friend who read it aloud at the Royal Society on 23 December 1763. Price edited Bayes's major work "An Essay Towards Solving a Problem in the Doctrine of Chances" (1763), which appeared in Philosophical Transactions, and contains Bayes' theorem. Price wrote an introduction to the paper that provides some of the philosophical basis of Bayesian statistics and chose one of the two solutions Bayes offered. In 1765, Price was elected a Fellow of the Royal Society in recognition of his work on Bayes's legacy. On 27 April, a letter sent to his friend Benjamin Franklin was read out at the Royal Society, and later published, in which Price applies this work to population and computing 'life-annuities'.

Independently of Bayes, Pierre-Simon Laplace used conditional probability to formulate the relation of an updated posterior probability from a prior probability, given evidence. He reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work, and summarized his results in Théorie analytique des probabilités (1812). The Bayesian interpretation of probability was developed mainly by Laplace.

About 200 years later, Sir Harold Jeffreys put Bayes's algorithm and Laplace's formulation on an axiomatic basis, writing in a 1973 book that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry".

Stephen Stigler used a Bayesian argument to conclude that Bayes' theorem was discovered by Nicholas Saunderson, a blind English mathematician, some time before Bayes, but that is disputed.

Martyn Hooper and Sharon McGrayne have argued that Richard Price's contribution was substantial: