Bayes' rule

In probability theory and applications, Bayes's rule relates the odds of event to the odds of event , before (prior to) and after (posterior to) conditioning on another event . The odds on to event is simply the ratio of the probabilities of the two events. The prior odds is the ratio of the unconditional or prior probabilities, the posterior odds is the ratio of conditional or posterior probabilities given the event . The relationship is expressed in terms of the likelihood ratio or Bayes factor, . By definition, this is the ratio of the conditional probabilities of the event given that is the case or that is the case, respectively. The rule simply states: posterior odds equals prior odds times Bayes factor (Gelman et al., 2005, Chapter 1).