Valid uncertainty quantification about the model in a linear regression setting

TL;DR

This paper introduces an inferential model approach for valid, prior-free probabilistic inference in linear regression, enabling reliable uncertainty quantification across multiple model assertions.

Contribution

It develops an optimal IM framework for simultaneous valid inference without priors, connecting it to post-selection inference in linear regression.

Findings

Guarantees valid probabilistic inference over multiple assertions.

Achieves optimal efficiency among valid inference methods.

Provides a new approach to uncertainty quantification in model selection.

Abstract

In scientific applications, there often are several competing models that could be fit to the observed data, so quantification of the model uncertainty is of fundamental importance. In this paper, we develop an inferential model (IM) approach for simultaneously valid probabilistic inference over a collection of assertions of interest without requiring any prior input. Our construction guarantees that the approach is optimal in the sense that it is the most efficient among those which are valid. Connections between the IM's simultaneous validity and post-selection inference are also made. We apply the general results to obtain valid uncertainty quantification about the set of predictor variables to be included in a linear regression model.