In Bayesian probability theory, if, given a likelihood function
, the posterior distribution is in the same probability distribution family as the prior probability distribution , 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 .
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]
^Howard Raiffa and Robert Schlaifer. Applied Statistical Decision Theory. Division of Research, Graduate School of Business Administration, Harvard University, 1961.
^Jeff Miller et al. Earliest Known Uses of Some of the Words of Mathematics, "conjugate prior distributions". Electronic document, revision of November 13, 2005, retrieved December 2, 2005.
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