Bayes' Rule

In probability theory and applications, Bayes' rule relates the odds of event to event, before and after conditioning on event . The relationship is expressed in terms of the Bayes factor, . Bayes' rule is derived from and closely related to Bayes' theorem. Bayes' rule may be preferred to Bayes' theorem when the relative probability (that is, the odds) of two events matters, but the individual probabilities do not. This is because in Bayes' rule, is eliminated and need not be calculated (see Derivation). It is commonly used in science and engineering, notably for model selection.

Under the frequentist interpretation of probability, Bayes' rule is a general relationship between and, for any events, and in the same event space. In this case, represents the impact of the conditioning on the odds.

Under the Bayesian interpretation of probability, Bayes' rule relates the odds on probability models and before and after evidence is observed. In this case, represents the impact of the evidence on the odds. This is a form of Bayesian inference - the quantity is called the prior odds, and the posterior odds. By analogy to the prior and posterior probability terms in Bayes' theorem, Bayes' rule can be seen as Bayes' theorem in odds form. For more detail on the application of Bayes' rule under the Bayesian interpretation of probability, see Bayesian model selection.