Editing Bayes' theorem (section)

{{Short description|Mathematical rule for inverting probabilities}}
{{Redirect|Bayes rule|the concept in decision theory|Bayes estimator}}
{{Bayesian statistics}}

'''Bayes' theorem''' (alternatively '''Bayes' law''' or '''Bayes' rule''', after [[Thomas Bayes]]) gives a mathematical rule for inverting [[Conditional probability|conditional probabilities]], allowing one to find the probability of a cause given its effect. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to someone of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the person is typical of the population as a whole. Based on Bayes' law, both the prevalence of a disease in a given population and the error rate of an infectious disease test must be taken into account to evaluate the meaning of a positive test result and avoid the ''[[base-rate fallacy]]''.

One of Bayes' theorem's many applications is [[Bayesian inference]], an approach to [[statistical inference]], where it is used to invert the probability of [[Realization (probability)|observations]] given a model configuration (i.e., the [[likelihood function]]) to obtain the probability of the model configuration given the observations (i.e., the [[posterior probability]]).