where $Z = \{X, A, M\}$ includes pre-treatment measures $X$, treatment $A$, and post-treatment pre-outcome mediators $M$. Let $p(Y(\pi, a, a'))$ denote the potential outcome distribution that corresponds to the effect of $A$ on $Y$ along proper causal paths in $\pi$, where $\pi$ includes the direct influence of $A$ on $Y$, and let $p(Y_0(\pi, a, a'))$ denote the identifying functional for $p(Y(\pi, a, a'))$ obtained from the edge-formula in (\[eqn:pse\]), where the term $p(Y | Z)$ is evaluated at $\{Z \setminus A\} = 0$. Then ${\mathbb{E}}[Y | Z]$ can be written as follows: [ $$\begin{aligned}
        {\mathbb{E}}[Y | Z] = f(Z) -  \big( {\mathbb{E}}[Y(\pi, a, a')] - {\mathbb{E}}[Y_0(\pi, a, a')]\big) \  + \ \phi(A), 
            \end{aligned}$$ ]{} where $f(Z) := {\mathbb{E}}[Y | Z] - {\mathbb{E}}[Y | A, \{Z \setminus A\} = 0]$ and $\phi(A) = w_0 + w_aA$. Furthermore, $w_a$ corresponds to $\pi$-specific effect of $A$ on $Y$. \[theorem:reparam\]

To illustrate the above reparameterization, consider the graph in Fig. \[fig:graphs\](b), discussed in [@NabiShpitser18Fair] and [@chiappa2018path]. Assume the direct path and the paths through $M$ of $A$ on $Y$ are the impermissible pathways (depicted with green edges). The corresponding PSE is encoded by a counterfactual contrast with respect to [$Y(a, M(a), L(a', M(a)))$]{}. 