is not feasible. In next section, we explain how $p(X)$ can be incorporated into the constrained optimization problem using empirical likelihood methods.

Imposing Fairness Constraints With Hybrid Likelihoods {#sec:hybrid}
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In light of Theorem \[lem:KLdis\], we are interested in constraining the nonparameteric form of $p(X)$. Following work in [@OwenEL], we use hybrid/semi-parametric empirical likelihood methods to estimate $p(X)$ nonparametrically which is a novel idea in the fairness setting. First, according to Theorem \[lem:KLdis\], constraining $p(X)$ would bring our learned distribution closer to the observed (unfair) distribution, and hence results in improvement of model performance, as we demonstrate in our simulations. Second, $p(X)$ is often a high dimensional object that is difficult to estimate due to the curse of dimensionality. For simplicity of presentation, we focus on the DAG in Fig. \[fig:graphs\](a), and the constraint represented by the NDE, although the methods we describe generalize without difficulty to arbitrary causal models and constraints represented by arbitrary PSEs.

Let $(X_i, A_i, M_i, Y_i), i = 1, \ldots, n$ be independent random vectors with common distribution $p = p(X) \times p(A,M,Y | X)$. We assume a known parametric form for $p(A,M,Y | X)$, and leave $p(X)$ unrestricted. Assuming the unfair effect is NDE, 