Skip to main content
Springer Nature Link
Log in

Bayes’ Theorem

  • Reference work entry
  • First Online:
International Encyclopedia of Statistical Science

  • 1014 Accesses

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...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References and Further Reading

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Carnegie Mellon University, Pittsburg, PA, USA

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

Authors
  1. Joseph B. Kadane
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

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

    Miodrag Lovric

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this entry

Cite this entry

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

Download citation

Publish with us

Policies and ethics