The BGLIMM Procedure

Conjugate prior distributions are a type of prior distribution in which the prior and posterior distributions are in the same family of distributions. For example, if you model an independently and identically distributed random variable by using a normal likelihood with known variance ,

then a normal prior on ,

is a conjugate prior, because the posterior distribution of is also a normal distribution, where the covariance is and the mean is

PROC BGLIMM uses conjugate samplers in the normal and multivariate normal cases, as shown in Table 14.

Table 14: Conjugate Sampling in PROC BGLIMM

Family	Parameter	Prior
Normal		Multivariate normal
Normal	Variance	Inverse gamma
Multivariate normal		Multivariate normal
Multivariate normal	Covariance	Inverse Wishart

The fixed-effects parameters and the covariances and are sampled when applicable.