Priors

Conjugate Priors

Conjugate Prior

Suppose is a class of sampling distributions (likelihoods) , and is a class of prior distributions for , then the class is conjugate for if

Lemma Conjugation Property of Gaussian Suppose and is positive semidefinite symmetric matrix. Then

where , and . Proof

Prop Summary of Common Conjugate Priors

Prior	Likelihood	Posterior

Improper Priors

Def Improper Prior A prior distribution is improper if .

Def Sufficient We say that the empirical mean is sufficient for parameter if the posterior depends on the data only through , rather than the each .

Jeffreys’ Prior

Def Jeffreys’ Prior The Jeffreys’ prior is defined aswhere is the Fisher information.