We are very grateful to all discussants for their many insightful and inspiring comments. While we cannot respond in a brief rejoinder to every issue that has been raised, we present some additional thoughts relating to the stimulating contributions. Connections to Bayesian approaches. Richardson, Brown and Griffin, Draper and Leng, and Nott discuss interesting possible connections between stability selection (or other randomized selection procedures) and Bayesian approaches with appropriately chosen priors. Randomized Lasso has the most immediate relation, as pointed out by Brown and Griffin and connecting with their interesting paper (Griffin and Brown, 2007). They also raise the question whether subsampling is then still necessary. While we do not have a theoretical answer here, it seems that subsampling improves in practice a randomized procedure (or the equivalent Bayesian counterpart). We are also not “throwing away real data ” with subsampling since the final selection probabilities over subsampled data are U-statistics of order ⌊n/2 ⌋ and are using all n samples, not just a random subset. Stability selection is closely related to Bagging (Breiman, 1996), as pointed out by Richardson. Stability selection is aggregating selection outcomes rather than predictions and assigning an error rate via our Theorem 1. The paper Nott and Leng (2009) seems to be very interesting in the

In genome-wide association studies, penalization is becoming an important approach for identifying genetic markers associated with disease. Motivated by the fact that there exists natural grouping structure in SNPs and more importantly such groups are correlated, we propose a new penalization method for group variable selection which can properly accommodate the correlation between adjacent groups. This method is based on a combination of the group Lasso penalty and a quadratic penalty on difference of regression coefficients of adjacent groups. The new method is referred to as Smoothed Group Lasso, or SGL. It encourages group sparsity and smoothes regression coefficients for adjacent groups. Canonical correlations are applied to the weights between groups in the quadratic difference penalty. We derive a group coordinate descent algorithm for computing the solution path. This algorithm takes the solution of a closed form of SGL for a single group model and is efficient and stable in highdimensional settings. The SGL method is further extended to logistic regression for binary response. With the assistance of MM algorithm, the logistic regression model

Vol. 28 ISMB 2012, pages i137–i146 doi:10.1093/bioinformatics/bts227 Leveraging input and output structures for joint mapping of epistatic and marginal eQTLs