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

TL;DR

This paper introduces a two-stage projective inference approach for high-dimensional generalized linear models, balancing sparsity and prediction accuracy by constructing a reference model and then projecting onto a minimal feature subset.

Contribution

It proposes a novel projection technique unifying existing methods, along with a fast cross-validation method for feature selection and theoretical conditions for its effectiveness.

Findings

The new projection method is accurate and computationally efficient.

The approach effectively balances sparsity and predictive performance.

Empirical results demonstrate advantages on simulated and real data.

Abstract

This paper discusses predictive inference and feature selection for generalized linear models with scarce but high-dimensional data. We argue that in many cases one can benefit from a decision theoretically justified two-stage approach: first, construct a possibly non-sparse model that predicts well, and then find a minimal subset of features that characterize the predictions. The model built in the first step is referred to as the \emph{reference model} and the operation during the latter step as predictive \emph{projection}. The key characteristic of this approach is that it finds an excellent tradeoff between sparsity and predictive accuracy, and the gain comes from utilizing all available information including prior and that coming from the left out features. We review several methods that follow this principle and provide novel methodological contributions. We present a new projection technique that unifies two existing techniques and is both accurate and fast to compute. We also propose a way of evaluating the feature selection process using fast leave-one-out cross-validation that allows for easy and intuitive model size selection. Furthermore, we prove a theorem that helps to understand the conditions under which the projective approach could be beneficial. The benefits are illustrated via several simulated and real world examples.