  Causal inference relies on two fundamental assumptions: ignorability and
positivity. We study causal inference when the true confounder value can be
expressed as a function of the observed data; we call this setting estimation
with functional confounders (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 functional interventions and a
sufficient condition for effect estimation of these interventions called
functional positivity. 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.
