Editing Conjugate prior (section)

{{Short description|Concept in probability theory}}
{{Bayesian statistics}}

In [[Bayesian probability]] theory, if, given a [[likelihood function]] 
<math>p(x \mid \theta)</math>,  the [[posterior probability|posterior distribution]] <math>p(\theta \mid x)</math> is in the same [[List of probability distributions|probability distribution family]] as the [[prior probability distribution]] <math>p(\theta)</math>, the prior and posterior are then called '''conjugate distributions''' with respect to that  likelihood function and the prior is called a '''conjugate prior''' for the likelihood function <math>p(x \mid \theta)</math>. 

A conjugate prior is an algebraic convenience, giving a [[closed-form expression]] for the posterior; otherwise, [[numerical integration]] may be necessary. Further, conjugate priors may clarify how a likelihood function updates a prior distribution.

The concept, as well as the term "conjugate prior", were introduced by [[Howard Raiffa]] and [[Robert Schlaifer]] in their work on [[Bayesian decision theory]].<ref name="raiffa_schlaifer">[[Howard Raiffa]] and [[Robert Schlaifer]]. ''Applied Statistical Decision Theory''. Division of Research, Graduate School of Business Administration, Harvard University, 1961.</ref> A similar concept had been discovered independently by [[George Alfred Barnard]].<ref name="miller">Jeff Miller et al. [http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics], [http://jeff560.tripod.com/c.html "conjugate prior distributions"]. Electronic document, revision of November 13, 2005, retrieved December 2, 2005.</ref>