any $p(X)$ given a set of i.i.d draws is the empirical distribution which places mass $1/n$ at every observed point. Empirical likelihood methods have been developed for settings where the nonparametric and parametric (hybrid) likelihood must be maximized subject to moment constraints [@OwenEL]. We describe below how these methods may be adapted to our setting.

Finally, we show how both the reparameterization method and the empirical likelihood method can be combined to yield a constrained optimization method that maximizes a semi-parametric (hybrid reparameterized) likelihood using standard software.

EFFICIENT APPROXIMATION OF FAIR WORLDS {#sec:consMLE}
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Imposing Fairness Constraints With Reparameterized Likelihoods {#sec:likereparam}
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In this section, we describe how to reparameterize the observed data likelihood in terms of causal parameters that correspond to the effect of $A$ on $Y$ along certain causal pathways. The results presented in the following theorem greatly simplifies the constrained optimization problem shown in (\[eqn:c-mle\]) in settings where the PSE includes the direct influence of $A$ on $Y$. This is due to the fact that the constrained parameter, corresponding to the PSE of interest, now appears as a single coefficient in the outcome regression model.

Assume the observed data distribution $p(Y,Z)$ is induced by a *causal model*, 