# Bayes' theorem

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Bayes' theorem (also known as Bayes' rule or Bayes' law) is a result in probability theory that relates conditional probabilities. If A and B denote two events, P(A|B) denotes the conditional probability of A occurring, given that B occurs. The two conditional probabilities P(A|B) and P(B|A) are in general different. Bayes theorem gives a relation between P(A|B) and P(B|A).
An important application of Bayes' theorem is that it gives a rule how to update or revise the strengths of evidence-based beliefs in light of new evidence a posteriori.
As a formal theorem, Bayes' theorem is valid in all interpretations of probability. However, it plays a central role in the debate around the foundations of statistics: frequentist and Bayesian interpretations disagree about the kinds of things to which probabilities should be assigned in applications. Whereas frequentists assign probabilities to random events according to their frequencies of occurrence or to subsets of populations as proportions of the whole, Bayesians assign probabilities to propositions that are uncertain. A consequence is that Bayesians have more frequent occasion to use Bayes' theorem. The articles on Bayesian probability and frequentist probability discuss these debates at greater length.<section end=glossary />