  We consider the problem of variable selection in high-dimensional settings
with missing observations among the covariates. To address this relatively
understudied problem, we propose a new synergistic procedure -- adaptive
Bayesian SLOPE -- which effectively combines the SLOPE method (sorted $l_1$
regularization) together with the Spike-and-Slab LASSO method. We position our
approach within a Bayesian framework which allows for simultaneous variable
selection and parameter estimation, despite the missing values. As with the
Spike-and-Slab LASSO, the coefficients are regarded as arising from a
hierarchical model consisting of two groups: (1) the spike for the inactive and
(2) the slab for the active. However, instead of assigning independent spike
priors for each covariate, here we deploy a joint "SLOPE" spike prior which
takes into account the ordering of coefficient magnitudes in order to control
for false discoveries. Through extensive simulations, we demonstrate
satisfactory performance in terms of power, FDR and estimation bias under a
wide range of scenarios. Finally, we analyze a real dataset consisting of
patients from Paris hospitals who underwent a severe trauma, where we show
excellent performance in predicting platelet levels. Our methodology has been
implemented in C++ and wrapped into an R package ABSLOPE for public use.
