Conjugate Probability Distribution: Difference between revisions
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A [[Conjugate Probability Distribution]] is a [[probability distribution]] whose [[posterior probability|posterior | A [[Conjugate Probability Distribution]] is a [[probability distribution]] whose [[posterior probability|posterior distribution]]s ''p''(θ|''x'') is in the same family as its [[prior probability distribution]] ''p''(θ). | ||
* <B>AKA:</B> [[Conjugate Probability Distribution|Conjugate Families of Distributions]]. | * <B>AKA:</B> [[Conjugate Probability Distribution|Conjugate Families of Distributions]]. | ||
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=== 2014 === | === 2014 === | ||
* (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/conjugate_prior Retrieved:2014-8-24. | * (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/conjugate_prior Retrieved:2014-8-24. | ||
** In [[Bayesian probability]] theory, if the [[posterior probability|posterior | ** In [[Bayesian probability]] theory, if the [[posterior probability|posterior distribution]]s ''p''(θ|''x'') are in the same family as the [[prior probability distribution]] ''p''(θ), the prior and posterior are then called <B>conjugate distributions,</B> and the prior is called a <B>conjugate prior</B> for the [[likelihood function]]. For example, the [[Normal distribution|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. | ||
<references/> | <references/> | ||
Latest revision as of 07:28, 22 August 2024
A Conjugate Probability Distribution is a probability distribution whose posterior distributions p(θ|x) is in the same family as its prior probability distribution p(θ).
- AKA: Conjugate Families of Distributions.
- …
- Example(s):
- an Exponential Family.
- …
- Counter-Example(s):
- See: Bayesian Probability, Normal Distribution, Bayesian Decision Theory, Bayes' Theorem, Closed-Form Expression.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/conjugate_prior Retrieved:2014-8-24.
- 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.
2006
- (Gelman, 2006) ⇒ Andrew Gelman. (2006). “Prior Distributions for Variance Parameters in Hierarchical Models." Bayesian analysis, 1(3).
2005
- (DeGroot, 2005) ⇒ Morris H. DeGroot. (2005). “Conjugate Prior Distributions." Optimal Statistical Decisions.