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This paper discusses predictive inference and feature selection forgeneralized linear models with scarce but high-dimensional data. We argue thatin many cases one can benefit from a decision theoretically justified two-stageapproach: first, construct a possibly non-sparse model that predicts well, andthen find a minimal subset of features that characterize the predictions. Themodel built in the first step is referred to as the \emph{reference model} andthe operation during the latter step as predictive \emph{projection}. The keycharacteristic of this approach is that it finds an excellent tradeoff betweensparsity and predictive accuracy, and the gain comes from utilizing allavailable information including prior and that coming from the left outfeatures. We review several methods that follow this principle and providenovel methodological contributions. We present a new projection technique thatunifies two existing techniques and is both accurate and fast to compute. Wealso propose a way of evaluating the feature selection process using fastleave-one-out cross-validation that allows for easy and intuitive model sizeselection. Furthermore, we prove a theorem that helps to understand theconditions under which the projective approach could be beneficial. Thebenefits are illustrated via several simulated and real world examples.

Projective Inference in High-dimensional Problems: Prediction and Feature Selection