Exact Post Model Selection Inference for Marginal Screening

TL;DR

This paper introduces an exact, non-asymptotic framework for post-model selection inference in linear regression, enabling valid confidence intervals and hypothesis tests after marginal screening without eigenvalue assumptions.

Contribution

It provides a novel exact distribution characterization for linear functions of responses conditioned on model selection, applicable to various selection procedures.

Findings

Exact distribution results for post-selection inference

Framework applicable to multiple selection procedures

Computationally efficient for large datasets

Abstract

We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response $y$, conditional on the model being selected (``condition on selection" framework). This allows us to construct valid confidence intervals and hypothesis tests for regression coefficients that account for the selection procedure. In contrast to recent work in high-dimensional statistics, our results are exact (non-asymptotic) and require no eigenvalue-like assumptions on the design matrix $X$. Furthermore, the computational cost of marginal regression, constructing confidence intervals and hypothesis testing is negligible compared to the cost of linear regression, thus making our methods particularly suitable for extremely large datasets. Although we focus on marginal screening to illustrate the applicability of the condition on selection framework, this framework is much more broadly applicable. We show how to apply the proposed framework to several other selection procedures including orthogonal matching pursuit, non-negative least squares, and marginal screening+Lasso.