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Bayes' theorem (also known as Bayes' rule or Bayes' law) is a result in
probability theory, which relates the conditional and marginal probability
distributions of random variables. In some interpretations of probability,
Bayes' theorem tells how to update or revise beliefs in light of new evidence a
posteriori.
The probability of an event A conditional on another event B is generally
different from the probability of B conditional on A. However, there is a
definite relationship between the two, and Bayes' theorem is the statement of
that relationship.
As a formal theorem, Bayes' theorem is valid in all interpretations of
probability. However, frequentist and Bayesian interpretations disagree about
the kinds of things to which probabilities should be assigned in applications:
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.