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 learned. Bayesian updating is an important technique throughout statistics, and especially in mathematical statistics: Exhibiting a Bayesian derivation for a statistical method automatically ensures that the method works as well as any competing method, for some cases. 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, 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; however, non-Bayesian updating rules are compatible with rationality, according to philosophers Ian Hacking and Bas van Fraassen.
Read more about Bayesian Inference: Inference Over Exclusive and Exhaustive Possibilities, In Frequentist Statistics and Decision Theory, Bayes and Bayesian Inference, History
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“Rules and particular inferences alike are justified by being brought into agreement with each other. A rule is amended if it yields an inference we are unwilling to accept; an inference is rejected if it violates a rule we are unwilling to amend. The process of justification is the delicate one of making mutual adjustments between rules and accepted inferences; and in the agreement achieved lies the only justification needed for either.”
—Nelson Goodman (b. 1906)