Conjugate prior

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In Bayesian probability theory, if, given a likelihood function ${\displaystyle p(x\mid \theta )}$, the posterior distribution ${\displaystyle p(\theta \mid x)}$ is in the same probability distribution family as the prior probability distribution ${\displaystyle p(\theta )}$, 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 ${\displaystyle p(x\mid \theta )}$.

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]