Bayes' theorem

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Bayesian statistics
Posterior = Likelihood × Prior ÷ Evidence
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In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.^[1] For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the individual is typical of the population as a whole.

One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in the theorem may have different probability interpretations. With Bayesian probability interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of related evidence. Bayesian inference is fundamental to Bayesian statistics. It has been considered to be "to the theory of probability what Pythagoras's theorem is to geometry."^[2]

Based on Bayes law both the prevalence of a disease in a given population and the error rate of an infectious disease test have to be taken into account to evaluate the meaning of a positive test result correctly and avoid the base-rate fallacy.