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

Abstract

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 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.

Robust Lasso‐Zero for sparse corruption and model selection with missing covariates

“…presented a variant of orthogonal matching pursuit which recovers the support and achieves the minimax optimal rate. Jiang et al (2019) proposed Adaptive Bayesian SLOPE, combining SLOPE and Spike-and-Slab Lasso. While some of these methods have interesting theoretical guarantees, they all require an estimation of the design covariance matrix, which is often obtained under the restrictive MCAR (Missing Completely At Random) assumption, when the missingness does not depend on the data.…”