q(Z_1) = p(Z_1) \big\}$, and $M_2 = \big\{p^*_2(Z) = \operatorname*{arg\,max}_{q(Z)}   D_{KL}(p || q) \hspace{0.1cm} \text{ s.t. } \hspace{0.1cm} \epsilon_l \leq g(q(Z)) \leq \epsilon_u, \ q(Z_2) = p(Z_2) \big\}$. If $Z_2 \subseteq Z_1 \subseteq Z$, then $D_{KL}(p || p^*_2) \leq D_{KL}(p || p^*_1).$ \[lem:KLdis\]

In other words, if a larger part of the joint is being constrained in $M_2$ compared to $M_1$, then $p^*_2(Z)$ is at least as close to $p(Z)$ as $p^*_1(Z)$.

To address the first difficulty, we provide a novel reparameterization of the observed data likelihood such that the causal parameter corresponding to the unfair PSE appears directly in the likelihood. This approach generalizes previous work on reparameterizations implied by structural nested models [@robins99marginal; @tchetgen14semi] to apply to a wide class of PSEs. With such a reparameterization, the MLE with a constrained PSE simply corresponds to maximum likelihood inference in a submodel where a certain likelihood parameter is set to $0$. This type of inference can be implemented with standard software.

To address the second difficulty, we propose an approach to constraining the density $p(X)$. An alternative to fully parametric modeling is to consider nonparametric representations of $p(X)$. It is well known that the nonparametric maximum likelihood estimate of 