Conjugate Probability Distribution

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):
  • a Smoothness Prior.
• See: Bayesian Probability, Normal Distribution, Bayesian Decision Theory, Bayes' Theorem, Closed-Form Expression.

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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.

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