possible values are averaged out, weighted appropriately by how likely they are according to the estimated $p^*$. This entails predicting $Y$ as the expected value ${\mathbb{E}}^*[Y | Z_2]$ (with respect to the distribution $\sum_{{Z}_1} p^*(Y, Z_1 | Z_2))$.

Next, we explain some limitations of the inference procedure described here and present our main contributions to address these limitations.

FAIR PREDICTIVE MODELS IN A BATCH SETTING {#sec:fairpred}
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Prediction problems in machine learning are typically tackled from the perspective of nonparametric risk minimization and the “train-and-test” framework. Here, we instead take the perspective of maximum likelihood and missing data, i.e., we treat unknown outcomes as missing values which we hope to impute in a way that is consistent with our specified likelihood for the entire data set. Our motivation for doing so is the nature of our constrained prediction problem. Specifically, our causal constraints contain “nuisance” components (conditional expectations and conditional distributions derived from the observed data distribution) which must be modeled correctly to ensure the causal effects are reliably estimated. In the subsequent prediction step, we should predict in a way that is consistent with what has already been modeled – or else we fail to exploit all the information 