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In probability theory and statistics, Bayes's theorem (alternatively Bayes's law or Bayes's rule) is a result that is of importance in the mathematical manipulation of conditional probabilities. Bayes's rule can be derived from more basic axioms of probability, specifically conditional probability.
When applied, the probabilities involved in Bayes's theorem may have any of a number of probability interpretations. In one of these interpretations, the theorem is used directly as part of a particular approach to statistical inference. ln particular, with the Bayesian interpretation of probability, the theorem expresses how a subjective degree of belief should rationally change to account for evidence: this is Bayesian inference, which is fundamental to Bayesian statistics. However, Bayes's theorem has applications in a wide range of calculations involving probabilities, not just in Bayesian inference.
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In statistics, Bayesian inference is a method of inference in which Bayes' rule is used to update the probability estimate for a hypothesis as additional evidence is acquired. Bayesian updating is an important technique throughout statistics, and especially in mathematical statistics. For some cases, exhibiting a Bayesian derivation for a statistical method automatically ensures that the method works as well as any competing method. Bayesian updating is especially important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a range of fields including science, engineering, philosophy, medicine and law.
In the philosophy of decision theory, Bayesian inference is closely related to discussions of subjective probability, often called "Bayesian probability". Bayesian probability provides a rational method for updating beliefs.
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In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable.
Bayes' theorem calculates the renormalized pointwise product of the prior and the likelihood function, to produce the posterior probability distribution, which is the conditional distribution of the uncertain quantity given the data.
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