We assume throughout the paper that both the phenotype YN×1 and the genotypes XN×M are standardized with mean zero and variance one. We assume a linear model YN×1=XN×MβM×1+εN×1 X, β and ε are mutually independent. We also assume that β is a random effect and effects of different SNPs are independent. A key idea in the AnnoPred framework is to utilize functional annotation information to accurately estimate SNPs’ effect sizes. In order to achieve that, we first partition trait heritability by annotations using LD score regression [18]. Since genotypes are standardized, per-SNP heritability is defined as the variance of βi for the ith SNP, and is used to quantify SNP effect sizes. More specifically, assume there are K + 1 pre-defined annotation categories, denoted as S0, S1, …, SK with S0 representing the entire genome. Under an additive assumption for heritability in overlapped annotations, we have βi∼N(0,∑j:i∈Sjτj), where τ0, τ1, …, τK, quantify the contribution to per-SNP heritability from each annotation category. Denote the estimated marginal effect size of the ith SNP as β^i=XiTYN, then we have the following approximation
