which unfair path-specific effects appear directly as parameters. This allows us to greatly simplify the constrained optimization problem, which has previously required complex or inefficient algorithms. Second, we demonstrate how tools from the empirical likelihood literature [@OwenEL] can be readily adapted to construct hybrid (semi-parametric) observed data likelihoods that satisfy given fairness criteria. With this approach, the entire likelihood is constrained, rather than only part of the likelihood as in past proposals [@NabiShpitser18Fair]. As a result, we use the data more efficiently and achieve better performance. Finally, we show how both innovations may be combined into a single procedure.

As a guiding example, we consider a setting such as automated hiring, in which we want to predict job success from applicant data. We have historical data on job success, resumes, and demographics, as well as new individuals for which we only see resumes and demographics for whom we would like to estimate a risk score with our predictive model. This may be considered a variant of semi-supervised learning or prediction with missing labels on a subset of the population. We aim to estimate those scores subject to path-specific fairness constraints. In order to describe the various components of this proposal, 