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# Bayes' theorem

The Bayes' theorem (attributed to Rev. Thomas Bayes, c. 1701-1761) is a simple result in probability theory. Despite its simplicity, it can be used to derive very interesting and useful results by combining prior knowledge and empirical evidence.

The prior knowledge is represented in terms of a probability, Pr[hypothesis], associated to any given hypothesis. In the case of stemmatological analysis, the hypothesis may, for instance, correspond to a statement that a specific stemma represent the actual copying history of the tradition under study. In case there is no reason to prefer any of the stemmata over others, the prior may be uniform so that Pr[hypothesis] = 1/M, where M is the number of possible stemmata.

The application of Bayes' theorem requires that the probability of the empirical evidence given a hypothesis can be evaluated. If, for instance, evidence is in the form of a number of extant textual variants and the hypothesis corresponds to a stemma, we also need a probabilistic model describing the probabilities of different changes of the text in the process of copying. Given such a model, we can evaluate the probability of the evidence, Pr[evidence | hypothesis].

The theorem states that the prior probability, Pr[hypothesis], and the probability of the evidence given the hypothesis, Pr[evidence | hypothesis], can be combined in the following way to obtain the posterior probability of the hypothesis given the evidence, Pr[hypothesis | evidence]:

Pr[hypothesis | evidence] = Pr[hypothesis] x Pr[evidence | hypothesis] / Pr[evidence],

where 'x' denotes multiplication.

The last quantity, Pr[evidence], may sometimes cause difficulties since it requires that we are able to associate a probability to the evidence without an assumed hypothesis. This often requires that we evaluate the sum of Pr[hypothesis] x Pr[evidence | hypothesis] over all possible hypotheses, which may be computationally infeasible. However, even if this is not possible, it is possible to obtain relative posterior probabilities of different hypotheses from the derived formula:

Pr[hypothesis1 | evidence] / Pr[hypothesis2 | evidence] = Pr[hypothesis1] / Pr[hypothesis2] x Pr[evidence | hypothesis1] / Pr[evidence | hypothesis2],

where the problematic term Pr[evidence] does not appear.

Some controversy is associated to Bayes' theorem, which is related to the subjective interpretation of probability as a degree of belief, and its role as the basis of Bayesian statistics and more particularly, Bayesian phylogenetics. However, the theorem itself is a direct consequence of the basic axioms of probability and hence, its mathematical validity is under no controversy.

#### In other languages

DE: Satz von Bayes
FR: théorème de Bayes
IT: teorema di Bayes

TR, KHVM

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