The reparameterization in Theorem \[theorem:reparam\] amounts to: [$$\begin{aligned}
&{\mathbb{E}}[Y | Z] =  f(Z) \nonumber - \sum_{Z\setminus A} \big\{ f(Z) \times p(L | M, X, A=0) \times \nonumber \\ 
& \hspace{2cm} p(M | X, A = 1) \times p(X)\big\} + w_0  + {w_a} A,
\label{eq:exReparam}\end{aligned}$$ ]{}where $w_a$ represents the PSE of interest; see the appendix for more details. A special case of this reparameterization when $\pi$ includes only the direct edge $A \to Y$ is implicit in the work of [@tchetgen14semi].

Under linearity assumptions, the PSE of interest in Fig. \[fig:graphs\](b) has a simple form. Assume the data generating process in Fig. \[fig:graphs\](b) is the same as the one given in display (2) of [@chiappa2018path], where [${\text{PSE}} = \theta^y_a + \theta^y_m \theta^m_a + \theta^y_l \theta^l_m \theta^m_a$]{}. In this case, our reparameterization takes the following form: [ $$\begin{aligned}
    & {\mathbb{E}}[Y | X, A, M, L] = 
    \underbrace{\Big(\theta^y_x X + \theta^y_m M + \theta^y_l L\Big)}_{f(Z)} - \\ 
    & \underbrace{\Big( \big( \theta^m_0 \theta^y_m +  (\theta^l_0 + \theta^l_m\theta^m_0)\theta^y_l \big) + \big(\theta^y_m \theta^m_a + \theta^y_l \theta^l_m \theta^m_a \big) A \Big)}_{\sum_{Z\setminus A} \big\{f(Z) p(L | M, X, A = 0) p(M | X, A = 1) p(X)\big\}} + \\
    & \Big( \!\!\underbrace{\theta^y_0 \!+\! \big(\theta^m_0 \theta^y_m \!+\!  (\theta^l_0 