attributes on outcomes – a view which generalizes an example discussed in [@pearl09causality]. We provide a brief summary of their perspective on fairness in the following without defending it for lack of space; see @NabiShpitser18Fair for more details.

Consider an observed data distribution $p(Y,Z)$ induced by a causal model, where $Y$ is an outcome and $Z = \{X, A, M\}$ includes all baseline factors $X$, sensitive features $A$, and post-treatment pre-outcome mediators $M$. Context and background ethical considerations pick out some path-specific effect of the sensitive feature $A$ on the outcome $Y$ as unfair; we assume this effect is identified as a functional $g(p(Y,Z))$. Fix upper and lower bounds $\epsilon_l, \epsilon_u$ for the PSE, representing a tolerable range. The most relevant bounds in practice are $\epsilon_l = \epsilon_u = 0$ or approximately zero. @NabiShpitser18Fair propose to transform the inference problem on $p(Y,Z)$, the “unfair world,” into an inference problem on another distribution $p^*(Y,Z)$, called the “fair world,” which is close in the sense of minimal KL-divergence to $p(Y,Z)$ while also having the property that the PSE lies within $(\epsilon_l, \epsilon_u)$.

Given a dataset $\mathcal{D} = \{(Y_i,Z_i), i = 1, \ldots, n \}$ drawn from $p(Y,Z)$, a likelihood function $\cal{L}(\mathcal{D}; \alpha)$, 