identifiable missing not at random (MNAR) model. Let the random variable $R$ denote the missingness status of the outcome variable $Y$ for each instance. That is, $R=1$ for all rows in the historical data (since $Y$ is observed) and $R=0$ for all rows in the new instances. Then the observed data likelihood is $\prod_{i = 1}^{n=n_1+n_2} p(X_i, A_i, M_i) \ p(Y_i | X_i, A_i, M_i)^{R_i}$.

This likelihood function describes the probability of the entire data set, though only uses $Y$ values from historical data. We can then maximize the likelihood subject to the specified path-specific constraints, and associate predicted values $\hat{Y}_{new}$ to the new instances. Note that the setting where new instances arrive sequentially one-at-a-time is a special case of this general setup, which would require retraining on the full combined data after the arrival of each instance. Though this is computationally more intensive than the proposal in [@NabiShpitser18Fair] (where they only train once), it will deliver significantly more accurate predictions because it uses all available information. We will elaborate on this point in Section \[sec:consMLE\].

The approach to fair prediction outlined in [@NabiShpitser18Fair] suffers from two problems: one general and one specific to our setting here. First, their approach 