Bayesian inference

Bayesian inference (ˈbeɪziən or ˈbeɪʒən ) is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Bayes' theorem Bayesian probability Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem: where stands for any hypothesis whose probability may be affected by data (called evidence below). Often there are competing hypotheses, and the task is to determine which is the most probable. the prior probability, is the estimate of the probability of the hypothesis before the data , the current evidence, is observed. the evidence, corresponds to new data that were not used in computing the prior probability. the posterior probability, is the probability of given , i.e., after is observed. This is what we want to know: the probability of a hypothesis given the observed evidence. is the probability of observing given and is called the likelihood. As a function of with fixed, it indicates the compatibility of the evidence with the given hypothesis. The likelihood function is a function of the evidence, , while the posterior probability is a function of the hypothesis, . is sometimes termed the marginal likelihood or "model evidence".

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