To incorporate annotation information into the multiple GWAS model (7), similarly, we model the relationship between and as (8)where is the probability of a null SNP being annotated, is the probability of the first phenotype associated-SNP being annotated, is the probability of the second phenotype associated-SNP being annotated, and is the probability of jointly associated-SNP being annotated. Assuming the independence of SNP markers, the joint distribution can be written as (9)where and are the j-th row of and ; the second equation holds by assuming the independence between and , conditional on ; and the third equation holds by further assuming the independence between and for , conditional on .
