we must review some background on causal inference, path-specific effects, and constrained prediction.

CAUSAL INFERENCE AND A CAUSAL APPROACH TO FAIRNESS {#sec:prelim}
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Causal inference is concerned with quantities which describe the consequences of interventions. Causal models are often represented graphically, e.g. by directed acyclic graphs (DAGs). We will use capital letters ($V$) to denote sets of random variables as well as corresponding vertices in graphs and lowercase letters ($v$) to denote values or assignments to those random variables. A DAG consists of a set of vertices $V$ connected by directed edges ($V_i \to V_j$ for some $\{V_i,V_j\} \subseteq V$) such that there are no cycles. The set $\operatorname{pa}_{\cal G}(V_i) \equiv \{V_j \in V \mid V_j \to V_i \}$ denotes the parents of $V_i$ in DAG ${\cal G}$. ${\mathfrak X}_{A}$ denotes the statespace of $A \subseteq V$.

A causal model of a DAG ${\cal G}$ is a set of distributions defined on potential outcomes (a.k.a. counterfactuals). For example, we consider distributions $p(V(a))$ subject to some restrictions, where $V(a)$ represents the value of $V$ had all variables in $\operatorname{pa}_{\cal G}(V)$ been set, possibly contrary to fact, to value $a$. In this paper, we assume Pearl’s *functional model* [@pearl09causality] for a DAG 