Predictive Uncertainty Quantification with Missing Covariates

TL;DR

This paper addresses the challenge of quantifying predictive uncertainty with missing covariates, proposing a flexible framework that provides valid predictive sets conditioned on missing data patterns, supported by theoretical and experimental results.

Contribution

It introduces CP-MDA-Nested*, a novel method for predictive uncertainty quantification with missing covariates, under certain independence assumptions, and explores the fundamental limits of distribution-free approaches.

Findings

Predictive uncertainty increases with more missing data.

CP-MDA-Nested* provides valid predictive sets under independence assumptions.

Experimental results show promising performance beyond theoretical scope.

Abstract

Predictive uncertainty quantification is crucial in decision-making problems. We investigate how to adequately quantify predictive uncertainty with missing covariates. A bottleneck is that missing values induce heteroskedasticity on the response's predictive distribution given the observed covariates. Thus, we focus on building predictive sets for the response that are valid conditionally to the missing values pattern. We show that this goal is impossible to achieve informatively in a distribution-free fashion, and we propose useful restrictions on the distribution class. Motivated by these hardness results, we characterize how missing values and predictive uncertainty intertwine. Particularly, we rigorously formalize the idea that the more missing values, the higher the predictive uncertainty. Then, we introduce a generalized framework, coined CP-MDA-Nested*, outputting predictive sets in both regression and classification. Under independence between the missing value pattern and both the features and the response (an assumption justified by our hardness results), these predictive sets are valid conditionally to any pattern of missing values. Moreover, it provides great flexibility in the trade-off between statistical variability and efficiency. Finally, we experimentally assess the performances of CP-MDA-Nested* beyond its scope of theoretical validity, demonstrating promising outcomes in more challenging configurations than independence.