Model Selection Confidence Sets by Likelihood Ratio Testing

TL;DR

This paper introduces Model Selection Confidence Sets (MSCSs) based on likelihood ratio testing to address model selection ambiguity, providing a statistically valid set of models that contain the true model with high confidence.

Contribution

It extends confidence interval concepts to model selection, offering a method to identify all plausible models and measure variable importance under uncertainty.

Findings

MSCS guarantees asymptotic coverage of the true model.

The approach effectively captures model uncertainty in high-dimensional settings.

Numerical experiments demonstrate the method's practical utility.

Abstract

The traditional activity of model selection aims at discovering a single model superior to other candidate models. In the presence of pronounced noise, however, multiple models are often found to explain the same data equally well. To resolve this model selection ambiguity, we introduce the general approach of model selection confidence sets (MSCSs) based on likelihood ratio testing. A MSCS is defined as a list of models statistically indistinguishable from the true model at a user-specified level of confidence, which extends the familiar notion of confidence intervals to the model-selection framework. Our approach guarantees asymptotically correct coverage probability of the true model when both sample size and model dimension increase. We derive conditions under which the MSCS contains all the relevant information about the true model structure. In addition, we propose natural statistics based on the MSCS to measure importance of variables in a principled way that accounts for the overall model uncertainty. When the space of feasible models is large, MSCS is implemented by an adaptive stochastic search algorithm which samples MSCS models with high probability. The MSCS methodology is illustrated through numerical experiments on synthetic data and real data examples.