  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.
