Throughout this paper, we shall use to index SNPs, to index GWAS data sets, and to index the annotation data sets. We first consider the simplest case where we only have summary statistics (p-values) from just one GWAS data set, and then extend our model to handle multiple GWAS data sets and annotation data. Suppose we have performed hypothesis testing of genome-wide SNPs and obtained their p-values: (3)where is the number of SNPs. Consider the “two-groups model” [46], i.e., the obtained p-values are assumed to come from the mixture of null and non-null, with probability and , respectively. Let be the latent variables indicating whether the j-th SNP is null or non-null, where , , and , because a SNP can only be either null or non-null. Here means un-associated (null) and means associated (non-null). Then we have the following two-groups model: (4)where the p-values from the null group are from the Uniform distribution on [0,1], denoted as , and the p-values from the non-null group are from the Beta distribution with parameters (), where . We put the constraint
