On the consistency of supervised learning with missing values

TL;DR

This paper investigates the consistency of supervised learning methods with missing data, revealing that simple mean imputation can be consistent in certain cases and advocating for decision trees that directly handle missingness.

Contribution

It demonstrates the theoretical and empirical consistency of simple imputation methods and decision trees that incorporate missing values directly in supervised learning.

Findings

Mean imputation is consistent when missingness is non-informative.

Decision trees with missing values can achieve optimal prediction.

The 'missing incorporated in attribute' method effectively handles various missing data scenarios.

Abstract

In many application settings, the data have missing entries which make analysis challenging. An abundant literature addresses missing values in an inferential framework: estimating parameters and their variance from incomplete tables. Here, we consider supervised-learning settings: predicting a target when missing values appear in both training and testing data. We show the consistency of two approaches in prediction. A striking result is that the widely-used method of imputing with a constant, such as the mean prior to learning is consistent when missing values are not informative. This contrasts with inferential settings where mean imputation is pointed at for distorting the distribution of the data. That such a simple approach can be consistent is important in practice. We also show that a predictor suited for complete observations can predict optimally on incomplete data, through multiple imputation. Finally, to compare imputation with learning directly with a model that accounts for missing values, we analyze further decision trees. These can naturally tackle empirical risk minimization with missing values, due to their ability to handle the half-discrete nature of incomplete variables. After comparing theoretically and empirically different missing values strategies in trees, we recommend using the "missing incorporated in attribute" method as it can handle both non-informative and informative missing values.