In the first prior, we assumed the same proportion of causal SNPs but different effect sizes across annotation categories. We now describe the second prior that assumes different proportions of causal SNPs but the same effect size across annotation categories. To be specific, we assume the causal effect size to be Var(βcausal) = V, the total number of SNPs to be M0, and the overall proportion of causal SNPs to be p0. The total heritability H02 can then be written as H02=p0M0V. For the ith SNP, use Ti=(⋂j:i∈SjSj)∩(⋂k:i∉SkSkc) to denote the collection of SNPs that share the same annotation assignment with the ith SNP, and let MTi=|Ti|, i.e. the number of SNPs in the set. Then, the total heritability of SNPs in Ti is HTi2=pTiMTiV, with pTi denoting the proportion of causal SNPs in Ti. Following these notations, we have βi∼pTiN(0,V)+(1−pTi)δ0 where V=H0p0N0 and pTi=p0M0HTi2MTiH02. We use H^2 to estimate H02, and the following formula to estimate HTi2.
