we have already committed to in the constraint estimation step. We chose the maximum likelihood framework as the most natural and simplest approach to accomplish this. Alternative methods for coherently combining nuisance estimation with nonparametric risk minimization are left to future work.

Unlike [@NabiShpitser18Fair], we consider a batch prediction setting – this allows us to avoid the inefficient averaging described in the previous section. In our case, historical data (of sample size $n_1$) consists of observations on $\{X,A,M,Y\}$ and new instances (of size $n_2$) comprise a set of observations with just $\{X,A,M\}$. The outcome labels for new instances are missing data which we aim to predict, subject to fairness constraints. Instead of training our constrained model on historical data alone, we train on the combination of historical data and new instances. This seems complicated since the observed data likelihood for the combined data set includes some complete rows and some partially incomplete rows. However, we can borrow ideas from the literature on missing data to accomplish this task. Specifically, we can impute missing outcomes (“labels”) using appropriate functions of observed data. In this paper we assume the labels are *missing at random* (MAR) [@little2002statistical]. However, our methods extend to any 