BAYES' THEOREM

Bayes' theorem deals with the role of new information in revising probability estimates. The theorem assumes that the probability of a hypothesis (the posterior probability) is a function of new evidence (the likelihood) and previous knowledge (prior probability). The theorem is named after Thomas Bayes (1702–1761), a nonconformist minister who had an interest in mathematics. The basis of the theorem is contained in as essay published in the Philosophical Transactions of the Royal Society of London in 1763.

Bayes' theorem is a logical consequence of the product rule of probability, which is the probability (P) of two events (A and B) happening— P(A,B)—is equal to the conditional probability of one event occurring given that the other has already occurred—P(A|B)—multiplied by the probability of the other event happening—P(B). The derivation of the theorem is as follows: P(A,B) = P(A|B)×P(B) = P(B|A)×P(A)

Thus: P(A|B) = P(B|A)×P(A)/P(B).

Bayes' theorem has been frequently used in the areas of diagnostic testing and in the determination of genetic predisposition. For example, if one wants to know the probability that a person with a particular genetic profile (B) will develop a particular tumour type (A)—that is, P(A|B). Previous knowledge leads to the assumption that the probability that any individual will develop the specific tumour (P(A)) is 0.1 and the probability that an individual has the particular genetic profile (P(B)) is 0.2. New evidence establishes that the probability that an individual with the tumor—P(B|A)—has the genetic profile of interest is 0.5.

Thus: P(A|B) = 0.1×0.5/0.2 = 0.25

The adoption of Bayes' theorem has led to the development of Bayesian methods for data analysis. Bayesian methods have been defined as "the explicit use of external evidence in the design, monitoring, analysis, interpretation and reporting" of studies (Spiegelhalter, 1999). The Bayesian approach to data analysis allows consideration of all possible sources of evidence in the determination of the posterior probability of an event. It is argued that this approach has more relevance to decision making than classical statistical inference, as it focuses on the transformation from initial knowledge to final opinion rather than on providing the "correct" inference.

In addition to its practical use in probability analysis, Bayes' theorem can be used as a normative model to assess how well people use empirical information to update the probability that a hypothesis is true.

BAYES' THEOREM

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