\!+\! \theta^l_m\theta^m_0)\theta^y_l \big)}_{w_0} + \underbrace{\big(\theta^y_a \!+\! \theta^y_m \theta^m_a \!+\! \theta^y_l \theta^l_m \theta^m_a \big)}_{w_a \equiv \text{PSE}} \!A\!\Big)
    \end{aligned}$$ ]{}In order to move away from the linear setting and exploit more flexible techniques, [@chiappa2018path] makes assumptions on the latent variables. However, such assumptions are often hard to verify in practice. In contrast, our result is entirely nonparametric and does not rely on any assumptions beyond what is encoded in the causal DAG.

By Theorem \[theorem:reparam\], the constrained optimization problem in eq. (\[eqn:c-mle\]) simplifies significantly to the following optimization problem: [$$\begin{aligned}
    \widehat{\alpha} =  \arg \max_{\alpha} \hspace{0.2cm} {\cal L}_{Y, Z}({\cal D}; {\alpha})  \hspace{0.3cm} \text{subject to} \hspace{0.2cm} w_a = 0.\end{aligned}$$ ]{}In the prediction setting, i.e., finding optimal parameters for ${\mathbb{E}}[Y|Z; \alpha_y]$, this amounts to an unconstrained maximum likelihood problem with outcome regression taking the specific form: [$$\begin{aligned}
    &{\mathbb{E}}[Y | Z; { \alpha_y}] = \label{eq:reparam_NDE} \\
    &f(Z;  \alpha_f) -  \sum_{X, M} f(Z; \alpha_f) \ p(M | A = 0, X; {\alpha_m}) \ p(X) + w_0, 
    \nonumber \end{aligned}$$ ]{}where $f(Z) := {\mathbb{E}}[Y | Z] - {\mathbb{E}}[Y | A, X=0, M=0]$ and is parameterized by $\alpha_f$. In practice, for each $X_i$ in the data $p(X_i)$ is replaced with its empirical approximation $1/n$, since a parametric specification of $p(X)$ 