are specified parametrically. Though some combination of nonparametric risk minimization and empirical likelihood would be an interesting extension, how to accomplish this is an open question.

Imposing Fairness Constraints With Hybrid Reparameterized Likelihoods {#sec:unified}
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In Section \[sec:likereparam\], we reformulated the constrained optimization problem of interest by rewriting the likelihood in terms of the parameters we were interested in constraining, and directly setting those parameters to zero. However, we did not place any constraints on $p(X)$. In Section \[sec:hybrid\], we used hybrid likelihoods to constrain a nonparametric estimate of $p(X)$, but did not provide a convenient reparameterization of the likelihood in terms of relevant parameters. In this section we describe an approach to optimizing a *hybrid reparameterized likelihood* that combines the advantages of both proposals. This allows us to constrain the entire likelihood and do so with standard maximum likelihood software, since the constraint we must satisfy directly corresponds to a parameter in the hybrid likelihood.

For simplicity of presentation, we again focus on the constraining the NDE, although the methods we describe generalize without difficulty to arbitrary constraints represented by arbitrary PSEs. The direct effect can then be estimated by ${\mathbb{E}}_x[ m(X;  \alpha)]$, where $m(X;  \alpha)$ is given in 