requires solving a computationally challenging constrained optimization problem. Likelihood functions are not in general convex and the constraints on path-specific effects involve nonlinear and complicated functionals of the observed data distribution. This makes the proposed constrained optimization a daunting task that relies on complex optimization software (or computationally expensive methods such as rejection sampling), which does not always find high quality local optima. Second, @NabiShpitser18Fair propose to constrain only part of the likelihood. Specifically they do not constrain the density $p(X)$ over the baseline features (since this is high-dimensional and thus inplausible to model accurately in their parametric approach). The baseline density is instead estimated by placing $1/n$ mass at every observed data point. This is sub-optimal in the specific setting we consider, where we do not need to average over constrained variables. Constraining a larger part of the joint should lead to a fair world distribution KL-closer to the observed distribution, which leads to better predictive performance as long as the likelihood is correctly specified. This intuition is formalized in the following result:

Let $p(Z)$ denote the observed data distribution, $M_1 = \big\{p^*_1(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, \ 