Here we present a method of genome wide inferred study (GWIS) that provides an approximation of genome wide association study (GWAS) summary statistics for a variable that is a function of phenotypes for which GWAS summary statistics, phenotypic means and covariances are available. GWIS can be performed regardless of sample overlap between the GWAS of the phenotypes on which the function depends. As GWIS provides association estimates and their standard errors for each SNP, GWIS can form the basis for polygenic risk scoring, LD score regression, Mendelian randomization studies, biological annotation and other analyses. Here, we replicate a body mass index (BMI) GWAS using GWIS based on a height GWAS and a weight GWAS. We proceed to use a GWIS to further our understanding of the genetic architecture of schizophrenia and bipolar disorder.