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We consider the problem of variable selection in high-dimensional settingswith missing observations among the covariates. To address this relativelyunderstudied problem, we propose a new synergistic procedure -- adaptiveBayesian SLOPE -- which effectively combines the SLOPE method (sorted $l_1$regularization) together with the Spike-and-Slab LASSO method. We position ourapproach within a Bayesian framework which allows for simultaneous variableselection and parameter estimation, despite the missing values. As with theSpike-and-Slab LASSO, the coefficients are regarded as arising from ahierarchical model consisting of two groups: (1) the spike for the inactive and(2) the slab for the active. However, instead of assigning independent spikepriors for each covariate, here we deploy a joint "SLOPE" spike prior whichtakes into account the ordering of coefficient magnitudes in order to controlfor false discoveries. Through extensive simulations, we demonstratesatisfactory performance in terms of power, FDR and estimation bias under awide range of scenarios. Finally, we analyze a real dataset consisting ofpatients from Paris hospitals who underwent a severe trauma, where we showexcellent performance in predicting platelet levels. Our methodology has beenimplemented in C++ and wrapped into an R package ABSLOPE for public use.

Adaptive Bayesian SLOPE -- High-dimensional Model Selection with Missing Values