We consider data from a Mendelian randomization study on N participants. For each participant, indexed by i, we measure J genetic variants (Gi1,Gi2,…,GiJ), a modifiable exposure, (Xi) and an outcome (Yi). We assume that confounders (represented by a single variable Ui) are unknown. The genetic variants are assumed to take the values 0, 1 or 2 (representing the number of exposure-increasing alleles of a bi-allelic single nucleotide polymorphism). The exposure is taken as a linear function of the genetic variants, the confounders and an independent error term (ϵiX). The coefficients γj for each variant j represent the effects of the genetic variants on the exposure. The outcome is taken as a linear function of the genetic variants, the exposure, the confounders and an independent error term (ϵiY). The causal effect of the exposure on the outcome is β. The coefficients αj for each variant j represent the direct effects of the genetic variants on the outcome that are not mediated by the exposure. The total effect of each variant on the outcome comprises the direct effect (αj) and the indirect
