The BGLIMM Procedure

GLM Parameters

In generalized linear models that have only fixed effects, the log of the posterior density is

log left-parenthesis p left-parenthesis bold-italic beta vertical-bar bold y comma bold upper R right-parenthesis right-parenthesis equals log left-parenthesis pi left-parenthesis bold-italic beta right-parenthesis right-parenthesis plus sigma-summation Underscript i equals 1 Overscript n Endscripts log left-parenthesis f left-parenthesis bold y Subscript i Baseline vertical-bar bold-italic beta comma bold upper R right-parenthesis right-parenthesis

where bold-italic beta is the vector of fixed-effects parameters and log left-parenthesis pi left-parenthesis bold-italic beta right-parenthesis right-parenthesis is the log of the joint prior density of bold-italic beta. The log likelihood, log left-parenthesis f left-parenthesis bold y Subscript i Baseline vertical-bar bold-italic beta comma bold upper R right-parenthesis right-parenthesis, is computed for the ith observation. The summation reflects the assumption that all observations in the data set are independent. The response variable bold y Subscript i can be a vector, and bold upper R can be either a scalar or a covariance. The logarithm of the prior distribution of bold upper R is not included because it is constant with respect to bold-italic beta.

The objective function of bold upper R is similar to that of bold-italic beta:

log left-parenthesis p left-parenthesis bold upper R vertical-bar bold y comma bold-italic beta right-parenthesis right-parenthesis equals log left-parenthesis pi left-parenthesis bold upper R right-parenthesis right-parenthesis plus sigma-summation Underscript i equals 1 Overscript n Endscripts log left-parenthesis f left-parenthesis bold y Subscript i Baseline vertical-bar bold-italic beta comma bold upper R right-parenthesis right-parenthesis
Last updated: December 09, 2022