We note that when SNP j is available in multiple GWAS summary statistics, the continuous shrinkage prior is shared across populations (i.e., both ϕ and Ψj do not depend on k), enabling information sharing between summary statistics while allowing for varying SNP effect sizes across populations to retain modeling flexibility. More specifically, given the variance parameters σk2, ϕ and Ψj, and the marginal least squares estimates of the SNP effect sizes in population k, β^k=XkTyk/Nk, the posterior mean of βk is E[βk|β^k]=(Dk+Ψ−1)−1β^k, where Dk=XkTXk/Nk is the LD matrix for population k, and Ψ = diag{Ψ1, Ψ2, ⋯ ,ΨM} is a diagonal matrix (Supplementary Note). It can be seen that Ψ does not depend on k and thus the amount of shrinkage applied to each SNP is shared across populations. Meanwhile, population-specific LD patterns are explicitly modelled via the LD matrix Dk.
