  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.
