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- Conjugate_prior abstract "In Bayesian probability theory, if the posterior distributions p(θ|x) are in the same family as the prior probability distribution p(θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function. For example, the Gaussian family is conjugate to itself (or self-conjugate) with respect to a Gaussian likelihood function: if the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian. This means that the Gaussian distribution is a conjugate prior for the likelihood which is also Gaussian. The concept, as well as the term "conjugate prior", were introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory. A similar concept had been discovered independently by George Alfred Barnard.Consider the general problem of inferring a distribution for a parameter θ given some datum or data x. From Bayes' theorem, the posterior distribution is equal to the product of the likelihood function and prior , normalized (divided) by the probability of the data Let the likelihood function be considered fixed; the likelihood function is usually well-determined from a statement of the data-generating process. It is clear that different choices of the prior distribution p(θ) may make the integral more or less difficult to calculate, and the product p(x|θ) × p(θ) may take one algebraic form or another. For certain choices of the prior, the posterior has the same algebraic form as the prior (generally with different parameter values). Such a choice is a conjugate prior.A conjugate prior is an algebraic convenience, giving a closed-form expressionfor the posterior: otherwise a difficult numerical integration may be necessary. Further, conjugate priors may give intuition, by more transparently showing how a likelihood function updates a prior distribution.All members of the exponential family have conjugate priors. See Gelman et al. for a catalog.".
- Conjugate_prior wikiPageID "846412".
- Conjugate_prior wikiPageRevisionID "599327790".
- Conjugate_prior group "note".
- Conjugate_prior hasPhotoCollection Conjugate_prior.
- Conjugate_prior subject Category:Bayesian_statistics.
- Conjugate_prior subject Category:Conjugate_prior_distributions.
- Conjugate_prior type Abstraction100002137.
- Conjugate_prior type Arrangement105726596.
- Conjugate_prior type Cognition100023271.
- Conjugate_prior type ConjugatePriorDistributions.
- Conjugate_prior type Distribution105729036.
- Conjugate_prior type PsychologicalFeature100023100.
- Conjugate_prior type Structure105726345.
- Conjugate_prior comment "In Bayesian probability theory, if the posterior distributions p(θ|x) are in the same family as the prior probability distribution p(θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function.".
- Conjugate_prior label "Conjugate prior".
- Conjugate_prior label "Distribuzione a priori coniugata".
- Conjugate_prior label "Сопряжённое априорное распределение".
- Conjugate_prior sameAs Distribuzione_a_priori_coniugata.
- Conjugate_prior sameAs m.03gm05.
- Conjugate_prior sameAs Q3711784.
- Conjugate_prior sameAs Q3711784.
- Conjugate_prior sameAs Conjugate_prior.
- Conjugate_prior wasDerivedFrom Conjugate_prior?oldid=599327790.
- Conjugate_prior isPrimaryTopicOf Conjugate_prior.