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International Encyclopedia of Statistical Science pp 89–90Cite as

Bayes’ Theorem

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The conditional probability P{A∣B} of event A given event B, is commonly defined as follows:

$$P\{A\mid B\} = \frac{P\{AB\}} {P\{B\}}$$
(1)

provided P{B} > 0. Alternatively, (1) can be reexpressed as

$$P\{AB\} = P\{A\mid B\}P\{B\}.$$
(2)

The left-hand side of (2) is symmetric in A and B, while the right-hand side does not appear to be. Therefore we have

$$P\{A\mid B\}P\{B\} = P\{B\mid A\}P\{A\},$$
(3)

or

$$P\{A\mid B\} = \frac{P\{B\mid A\}P\{A\}} {P\{B\}},$$
(4)

which is the first form of Bayes’ Theorem.

Note that the event B can be reexpressed as

$$B = AB \cup \overline{A}B.$$
(5)

Because AB and \(\overline{A}B\) are disjoint, we have

$$P\{B\} = P\{AB\} + P\{\overline{A}B\} = P\{B\mid A\}P\{A\} + P\{B\mid \overline{A}\}P\{\overline{A}\}.$$
(6)

Substituting (6) into (5) yields the second form of Bayes’ Theorem:

$$P\{A\mid B\} = \frac{P\{B\mid A\}P\{A\}} {P\{B\mid A\}P\{A\} + P\{B\mid \overline{A}\}P\{\overline{A}\}}.$$
(7)

Finally, let A 1, A 2, …, A k be disjoint sets whose union is...

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References and Further Reading

  • Fisher R (1959) Statistical methods and scientific inference, 2nd edn. Oliver and Boyd, Edinburgh and London

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Authors and Affiliations

  1. Carnegie Mellon University, Pittsburg, PA, USA

    Joseph B. Kadane (Leonard J. Savage University Professor of Statistics, Emeritus)

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  1. Joseph B. Kadane
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Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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© 2011 Springer-Verlag Berlin Heidelberg

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Kadane, J.B. (2011). Bayes’ Theorem. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_141

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