Causal Estimation with Functional Confounders.

Causal inference relies on two fundamental assumptions: and . We study causal inference when the true confounder value can be expressed as a function of the observed data; we call this setting EFC. In this setting ignorability is satisfied, however positivity is violated, and causal inference is impossible in general. We consider two scenarios where causal effects are estimable. First, we discuss interventions on a part of the treatment called and a sufficient condition for effect estimation of these interventions called . Second, we develop conditions for nonparametric effect estimation based on the gradient fields of the functional confounder and the true outcome function. To estimate effects under these conditions, we develop Level-set Orthogonal Descent Estimation (LODE). Further, we prove error bounds on LODE's effect estimates, evaluate our methods on simulated and real data, and empirically demonstrate the value of EFC.