By Bayes’ rule, the posterior distribution of β is: f(β|β^,D^)∝f(β^|β,D^)f(β) where D^=1NXTX is the sample correlation matrix and β^=1NXTY is the marginal effect size estimates. Given β and D^, β^ follows a multivariate normal distribution asymptotically with the following mean and variance E(β^|β,D^)=1N[E(XTXβ|β,D^)+E(XTε|β,D^)]=D^β Var(β^|β,D^)=Var(1NXTε|β,D^)=1N(1−hg2)D^.
