in the functional causal model (nor weaker causal models) with a corresponding graphical criterion on $\pi$ and ${\cal G}(V)$ called the ‘recanting witness’ [@shpitser13cogsci; @shpitser15hierarchy]. Under some assumptions, PSEs are nonparametrically identified by means of the *edge g-formula* described in [@shpitser15hierarchy]. As an example, consider the DAG in Fig. \[fig:graphs\](b). The PSE of $A$ on $Y$ along the paths $\pi = \{A \rightarrow Y, A\rightarrow L \rightarrow Y \}$ is encoded by a counterfactual contrast of the form $Y(\pi, a, a') = Y(a, M(a'), L(a, M(a')))$. This counterfactual density is identified by the edge g-formula as follows: [ $$\begin{aligned}
        &p(Y(a, M(a'), L(a, M(a'))) = \nonumber \\ 
        & \sum_{X, M, L} \ p(Y \mid a, X, M) \ p(L \mid a, M, X) \ p(M \mid a', X) \ p(X).
    \end{aligned}$$ ]{}For more details on PSEs, see [@shpitser13cogsci] and[@shpitser18medid], and [@nabi18pathpolicy].

Algorithmic Fairness via Constraining Path-Specific Effects {#sec:causal_fair}
-----------------------------------------------------------

There has been a growing interest in the issue of fairness in machine learning [@pedreshi08discrimination; @feldman15certifying; @hardt16equality; @kamiran13quantifying; @corbett2017algorithmic; @jabbari16fair; @kusner17counterfactual; @zhang18fairness; @Zhang17causal]. In this paper, we adopt the causal notion of fairness described in [@NabiShpitser18Fair] and [@nabi19fairpolicy], where unfairness corresponds to the presence of undesirable or impermissble path-specific effects of sensitive 