In probability theory, Bayes' theorem (often called Bayes' law and named after Rev Thomas Bayes; IPA:/'beɪz/) shows how one conditional probability (such as the probability of a hypothesis given observed evidence) depends on its inverse (in this case, the probability of that evidence given the hypothesis). The theorem expresses the posterior probability (i.e. after evidence E is observed) of a hypothesis H in terms of the prior probabilities of H and E, and the probability of E given H. It implies that evidence has a stronger confirming effect if it was more unlikely before being observed.

As a formal theorem, Bayes' theorem is valid in all common interpretations of probability. However, frequentist and Bayesian interpretations disagree on how (and to what) probabilities are assigned. In the Bayesian interpretation, probabilities are rationally coherent degrees of belief, or a degree of belief in a proposition given a body of well-specified information.

Bayes' theorem can then be understood as specifying how an ideally rational person responds to evidence. In the frequentist interpretation, probabilities are the frequencies of occurrence of random events as proportions of a whole. Though his name has become associated with subjective probability, Bayes himself interpreted the theorem in an objective sense.

The theorem was given extra prominence by a theorem by physicist R. T. Cox which showed that any system of inference fitting certain requirements could be mapped onto probability. Bayes' Theorem has since found a wide variety of applications in science and engineering.

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