Conjugate prior

In Bayesian probability theory, if 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, and the prior is called a conjugate prior for the likelihood function ${\displaystyle p(x\mid \theta )}$.

Part of a series on
Bayesian statistics
Posterior = Likelihood × Prior ÷ Evidence
Background
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Principle of indifference Principle of maximum entropy
Model building
Weak prior ... Strong prior Conjugate prior Linear regression Empirical Bayes Hierarchical model
Posterior approximation
Markov chain Monte Carlo Laplace's approximation Integrated nested Laplace approximations Variational inference Approximate Bayesian computation
Estimators
Bayesian estimator Credible interval Maximum a posteriori estimation
Evidence approximation
Evidence lower bound Nested sampling
Model evaluation
Bayes factor Model averaging Posterior predictive
Mathematics portal
v t e

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]