However, D^ is usually non-invertible and has very high dimensions. We thus study the posterior distribution of a small chunk of β^ instead. Let β^b be the estimated marginal effect size of SNPs in a region b (e.g. a LD block) and the corresponding genotype matrix is Xb and sample correlation matrix is D^b. Then the conditional mean and variance of β^b are E(β^b|βb,D^b)=1N[E(XbTXβ|βb,D^b)+E(XbTε|βb,D^b)]=D^bβb Var(β^b|βb,D^b)=1N2var(XbTXbβb+XbT(X−bβ−b+ε)|βb,D^b)=1N2var(XbT(X−bβ−b+ε)|βb,D^b)=1N2XbTvar(X−bβ−b+ε|βb,D^b)Xb=1N(1−hb2)D^b where hb2=∑i∈bσi2 is the heritability of SNPs in region b, and X−b and β−b denote the genotype matrix and effect sizes of SNPs not in region b. The conditional distribution of βb is: f(βb|β^b,D^b)∝N(D^bβb,1N(1−hb2)D^b)∏i∈bf(βi)∝{N(D^bβb,1N(1−hb2)D^b)∏i∈b[p0N(0,σi2p0)+(1−p0)δ0],underthefirstpriorN(D^bβb,1N(1−hb2)D^b)∏i∈b[pTiN(0,V)+(1−pTi)δ0],underthesecondprior
