and because they are products of a seemingly impartial optimization process. However, predictive models are trained on data that reflects the structural inequities, historical disparities, and other imperfections of our society. Often data includes sensitive attributes (e.g., race, gender, age, disability status), or proxies for such attributes. A particular worry in the context of data-driven decision-making is “perpetuating injustice,” which occurs when unfair dependence between sensitive features and outcomes is maintained, introduced, or reinforced by automated tools.

We study how to construct fair predictive models by correcting for the unfair causal dependence of predicted outcomes on sensitive features. We work with the proposed fairness criteria in [@NabiShpitser18Fair], where the authors propose that fair prediction requires imposing hard constraints on the prediction problem in the form of restricting certain causal path-specific effects. Impermissible pathways are user-specified and context-specific, hence require input from policymakers, legal experts, or the general public. Some alternative but also causally-motivated constrained prediction methods are proposed in [@chiappa2018path; @Kusner17fair] and [@zhang18fairness]. For a survey and discussion of distinct fairness criteria (both causal and associative) see [@mitchell2018prediction].

We advance the state of the art in two ways. First, we give a novel reparameterization of the observed data likelihood in 