Conjugate prior

In <a href="/facts/Bayesian_probability/Cvn7WWfG">Bayesian probability</a> theory, if, given a <a href="/facts/Likelihood_function/1L8P4NBX">likelihood function</a>

p
 (
 x
 ∣
 θ
 )
 
 
 {\displaystyle p(x\mid \theta )}
 
, the <a href="/facts/Posterior_probability/fojsMD0D">posterior distribution</a> 
 
 
 
 p
 (
 θ
 ∣
 x
 )
 
 
 {\displaystyle p(\theta \mid x)}
 
 is in the same <a href="/facts/List_of_probability_distributions/qrDTbPkd">probability distribution family</a> as the <a href="/facts/Prior_probability_distribution/JQKAD4o0">prior probability distribution</a> 
 
 
 
 p
 (
 θ
 )
 
 
 {\displaystyle p(\theta )}
 
, 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 
 
 
 
 p
 (
 x
 ∣
 θ
 )
 
 
 {\displaystyle p(x\mid \theta )}
 
. 
A conjugate prior is an algebraic convenience, giving a <a href="/facts/Closed-form_expression/y0xCXlSk">closed-form expression</a> for the posterior; otherwise, <a href="/facts/Numerical_integration/MwJnvcDV">numerical integration</a> may be necessary. Further, conjugate priors may give intuition by more transparently showing how a likelihood function updates a prior distribution.
The concept, as well as the term "conjugate prior", were introduced by <a href="/facts/Howard_Raiffa/HSPQtv7y">Howard Raiffa</a> and <a href="/facts/Robert_Schlaifer/lBFsVUEA">Robert Schlaifer</a> in their work on <a href="/facts/Bayesian_decision_theory/BpFRxYBg">Bayesian decision theory</a>. A similar concept had been discovered independently by <a href="/facts/George_Alfred_Barnard/LBhUL7IA">George Alfred Barnard</a>.

Conjugate prior open-in-new

Conjugate prior