PRS-CSx is an extension of PRS-CS19, which enables the integration of GWAS summary statistics from multiple populations to improve cross-population polygenic prediction. Consider the following Bayesian high-dimensional linear regression model for K populations: yk=Xkβk+ϵk, ϵk~MVN(0,σk2I), π(σk2)∝σk−2, k=1,2,⋯,K, where, for each population k, yk is a vector of standardized phenotypes (zero mean and unit variance) from Nk individuals, Xk is an Nk × Mk matrix of standardized genotypes (each column has zero mean and unit variance), βk is a vector of SNP effect sizes, ϵk is a vector of normally distributed non-genetic effects with variance σk2, for which we assign a non-informative scale-invariant Jeffreys prior, and I is an identify matrix. We use j = 1, 2, ⋯ , M to index the M unique SNPs across populations. For SNP j in population k, we place a continuous shrinkage prior on its effect size βjk, which can be represented as global-local scale mixtures of normals: βjk~N(0,σk2Nkψj), ψj~Gamma(a,δj), δj~Gamma(b,ϕ), where ϕ is a global shrinkage parameter shared across all SNPs that models the overall sparseness of the genetic architecture, and Ψj is
