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Bayes' theorem

Bayes' theorem, also known as Bayes' rule, is a result in probability theory, named after Thomas Bayes, who proved a special case of it. It is used in statistical inference to update estimates of the probability that different hypotheses are true, based on observations and a knowledge of how likely those observations are, given each hypothesis. In fact, it is habitually used by scientists in preference to the principle of induction. Bayes's theorem says that if an instance X is actually observed, then the probability of a hypothesis H must be multiplied by the following ratio:

probability of observing X if H is true
probability of observing X

In other words, the probability of a hypothesis H conditional on a given body of data X is equal the ratio of the unconditional probability of the conjunction of the hypothesis with the data to the unconditional probability of the data alone.

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