an estimator $\widehat{g}(\mathcal{D})$ of the unfair PSE, and bounds $\epsilon_l, \epsilon_u$, @NabiShpitser18Fair suggest to approximate $p^*(Y,Z)$ by solving the following constrained maximum likelihood problem: [$$\begin{aligned}
    &\widehat{\alpha} =  \arg \max_{\alpha} \hspace{0.2cm} {\cal L}_{Y, Z}({\cal D}; {\alpha}),  \hspace{0.3cm} \nonumber \\
    &\text{subject to} \hspace{0.2cm} \epsilon_l \leq \widehat{g}({\cal D}) \leq \epsilon_u.
    \label{eqn:c-mle}\end{aligned}$$ ]{}Having approximated the fair world $p^*(Y,Z; \widehat{\alpha})$ in this way, @NabiShpitser18Fair point out a key difficulty for using these estimated parameters to predict outcomes for new instances (e.g., new job applicants). A new set of observations $Z$ is not sampled from the “fair world" $p^*(Z)$ but from “unfair world" $p(Z)$. @NabiShpitser18Fair propose to map new instances from $p$ to $p^*$ and to use the result for predicting $Y$ with model parameters $\widehat{\alpha}$. They assume $Z$ can be partitioned into $Z_1$ and $Z_2$ such that $p^*(Y, Z) = p^*(Y, Z_1 | Z_2) p(Z_2)$. In other words, variables in $Z_2$ are shared between $p$ and $p^*$: $p^*(Z_2) = p(Z_2)$ but $p^*(Z_1 | Z_2) \neq p(Z_1 | Z_2)$. $Z_1$ typically corresponds to variables that appear in the estimator $\widehat{g}({\cal D})$. There is no obvious principled way of knowing exactly what values of $Z_1$ the “fair version" of the new instance would attain. Consequently, all such 