On the consistency of supervised learning with missing values

Abstract

In many application settings, data have missing entries, which makes subsequent analyses challenging. An abundant literature addresses missing values in an inferential framework, aiming at 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 test data. We first rewrite classic missing values results for this setting. We then show the consistency of two approaches, test-time multiple imputation and single imputation in prediction. A striking result is that the widely-used method of imputing with a constant prior to learning is consistent when missing values are not informative. This contrasts with inferential settings where mean imputation is frowned upon as it distorts the distribution of the data. The consistency of such a popular simple approach is important in practice. Finally, to contrast procedures based on imputation prior to learning with procedures that optimize the missing-value handling for prediction, we consider decision trees. Indeed, decision trees are among the few methods that can tackle empirical risk minimization with missing values, due to their ability to handle the half-discrete nature of incomplete variables. After comparing 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.