Phase transition and uncertainty quantification in variable selection and multiple testing

We study the problem of identifying the set of \emph{active} variables, termed in the literature as \emph{variable selection} or \emph{multiple hypothesis testing}, depending on the pursued criteria. For a general \emph{distribution-free} setting of non-normal, possibly dependent observations and a generalized notion of \emph{active set}, we propose a procedure that is used simultaneously for the both tasks, variable selection and multiple testing. The procedure is based on the \emph{risk hull minimization} method, but can also be obtained as a result of an empirical Bayes approach or a penalization strategy. We address its quality via various criteria: the Hamming risk, FDR, FPR, FWER, NDR, FNR, and various \emph{multiple testing risks}, e.g., MTR=FDR+NDR, and exhibit the peculiar \emph{phase transition} phenomenon. Finally, we introduce and study, for the first time, the \emph{uncertainty quantification} problem in the variable selection and multiple testing context.