We assume that the marginal effect size of each SNP is drawn from the following mixture distribution: βj~{N(0,τ2)withprobabilityπ0otherwise, where τ2 is the effect-size variance for non-null SNPs and π is the fraction of non-null SNPs in our data. We estimate the parameters τ2 and π by maximum likelihood. Given their values, the posterior distribution of SNP j can be calculated from Bayes’ Rule. Relative to the GWAS effect estimate, the mean of the posterior distribution is shrunken toward zero (because zero is the mean of the prior distribution) and is not biased by the winner’s curse. Further details and a derivation of the likelihood function used in the maximum-likelihood estimation are provided on p. 59 in the Supplementary Note of a previous SSGAC study37.
