# Conjugate prior

## Concept in probability theory / From Wikipedia, the free encyclopedia

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In Bayesian probability theory, if the posterior distribution $p(\theta \mid x)$ is in the same probability distribution family as the prior probability distribution $p(\theta )$, the prior and posterior are then called **conjugate distributions,** and the prior is called a **conjugate prior** for the likelihood function $p(x\mid \theta )$.

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Posterior = Likelihood × Prior ÷ Evidence |

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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 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 Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory.[1] A similar concept had been discovered independently by George Alfred Barnard.[2]