Bayes theorem

Bayes' theorem is used in probability theory, particularly conditional and marginal probability distributions of variables. Bayes' theorem guides the revision of probability estimates in view of new evidence. The probability of event A may depend on event B but that relationship may not be reciprocal: the probability of B may not be conditional on A, but there is an equation that connects the two, and Bayes' theorem is that equation: where L(A|B) is a likelihood of A given B for a fixed value of B. Each term in Bayes' theorem has a conventional name: Pr(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it does not take into account any information about B. Pr(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B. Pr(B|A) is the conditional probability of B given A. Pr(B) is the prior or marginal probability of B, and acts as a normalizing constant. With this terminology, the theorem may be paraphrased as In addition, the ratio Pr(B|A)/Pr(B) is sometimes called the standardized likelihood, so the theorem may also be paraphrased 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. “