Valid and efficient possibilistic structure learning in Gaussian linear regression

TL;DR

This paper introduces a new possibilistic inferential model framework for Gaussian linear regression that reliably quantifies uncertainty in model structure selection, even with incomplete prior information, ensuring finite-sample coverage.

Contribution

It extends the possibility-theoretic IM framework to include regularization with incomplete prior knowledge, providing calibrated, finite-sample valid confidence sets for model structure.

Findings

The new method achieves nominal coverage probability in finite samples.

It effectively incorporates incomplete prior information for structure regularization.

Benchmark data analysis shows improved uncertainty quantification over existing methods.

Abstract

A crucial step in fitting a regression model to data is determining the model's structure, i.e., the subset of explanatory variables to be included. However, the uncertainty in this step is often overlooked due to a lack of satisfactory methods. Frequentists have no broadly applicable confidence set constructions for a model's structure, and Bayesian posterior credible sets do not achieve the desired finite-sample coverage. In this paper, we propose an extension of the possibility-theoretic inferential model (IM) framework that offers reliable, data-driven uncertainty quantification about the unknown model structure. This particular extension allows for the inclusion of incomplete prior information about the unknown structure that facilitates regularization. We prove that this new, regularized, possibilistic IM's uncertainty quantification is suitably calibrated relative to the set of joint distributions compatible with the data-generating process and assumed partial prior knowledge about the structure. This implies, among other things, that the derived confidence sets for the unknown model structure attain the nominal coverage probability in finite samples. We provide background and guidance on quantifying prior knowledge in this new context and analyze two benchmark data sets, comparing our results to those obtained by existing methods.