What's a good imputation to predict with missing values?

TL;DR

This paper demonstrates that for predicting with missing data, an impute-then-regress approach with a powerful learner is theoretically optimal across all missing data mechanisms, and proposes a joint imputation-regression neural network method that outperforms traditional two-step procedures.

Contribution

It provides a theoretical foundation showing the optimality of impute-then-regress, and introduces NeuMiss, a neural network for joint imputation and regression that improves prediction accuracy.

Findings

Impute-then-regress is Bayes optimal for almost all imputation functions.

Perfect conditional imputation is not necessary for good asymptotic prediction.

NeuMiss outperforms traditional two-step imputation and regression methods in experiments.

Abstract

How to learn a good predictor on data with missing values? Most efforts focus on first imputing as well as possible and second learning on the completed data to predict the outcome. Yet, this widespread practice has no theoretical grounding. Here we show that for almost all imputation functions, an impute-then-regress procedure with a powerful learner is Bayes optimal. This result holds for all missing-values mechanisms, in contrast with the classic statistical results that require missing-at-random settings to use imputation in probabilistic modeling. Moreover, it implies that perfect conditional imputation is not needed for good prediction asymptotically. In fact, we show that on perfectly imputed data the best regression function will generally be discontinuous, which makes it hard to learn. Crafting instead the imputation so as to leave the regression function unchanged simply shifts the problem to learning discontinuous imputations. Rather, we suggest that it is easier to learn imputation and regression jointly. We propose such a procedure, adapting NeuMiss, a neural network capturing the conditional links across observed and unobserved variables whatever the missing-value pattern. Experiments confirm that joint imputation and regression through NeuMiss is better than various two step procedures in our experiments with finite number of samples.