Fast Marginal Likelihood Estimation of Penalties for Group-Adaptive Elastic Net

Abstract

Elastic net penalization is widely used in high-dimensional prediction and variable selection settings. Auxiliary information on the variables, for example, groups of variables, is often available. Group-adaptive elastic net penalization exploits this information to potentially improve performance by estimating group penalties, thereby penalizing important groups of variables less than other groups. Estimating these group penalties is, however, hard due to the high dimension of the data. Existing methods are computationally expensive or not generic in the type of response. Here we present a fast method for estimation of group-adaptive elastic net penalties for generalized linear models. We first derive a low-dimensional representation of the Taylor approximation of the marginal likelihood for group-adaptive ridge penalties, to efficiently estimate these penalties. Then we show by using asymptotic normality of the linear predictors that this marginal likelihood approximates that of elastic net models. The ridge group penalties are then transformed to elastic net group penalties by matching the ridge prior variance to the elastic net prior variance as function of the group penalties. The method allows for overlapping groups and unpenalized variables, and is easily extended to other penalties. For a model-based simulation study and two cancer genomics applications we demonstrate a substantially decreased computation time and improved or matching performance compared to other methods. Supplementary materials for this article are available online.