If we consider the set of the “reasons” that an event occurs, Bayes' theorem gives a formula for the probability that the event is the direct result of a particular reason.
Therefore, Bayes' theorem can be interpreted as a formula for the conditional probability of an event.
HISTORY
Bayes' theorem is named after Bayes, Thomas, and was developed in the middle of eighteenth century. However, Bayes did not publish the theorem during his lifetime; instead, it was presented by Price, R. on the 23rd December 1763, two years after his death, to the Royal Society of London, which Bayes was a member of during the last twenty last years of his life.
MATHEMATICAL ASPECTS
Let \( { \{ A_1,A_2,\ldots,A_k \} } \) be a partition of the sample space Ω. We suppose that each event \( { A_1, \ldots, A_k } \) has a nonzero probability. Let E be an event such that \( { P (E) > 0 } \).
So, for every \( { i (1 \leq i \leq k) } \), Bayes' theorem (for the discrete case) gives:
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REFERENCE
Bayes, T.: An essay towards solving a problem in the doctrine of chances. Philos. Trans. Roy. Soc. Lond. 53, 370–418 (1763). Published, by the instigation of Price, R., 2 years after his death. Republished with a biography by Barnard, George A. in 1958 and in Pearson, E.S., Kendall, M.G.: Studies in the History of Statistics and Probability. Griffin, London, pp. 131–153 (1970)
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(2008). Bayes' Theorem. In: The Concise Encyclopedia of Statistics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-32833-1_21
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