Causal Estimation with Functional Confounders
This work addresses a specific challenge in causal inference for researchers dealing with functional confounders, offering theoretical conditions and a new estimation method, but it is incremental as it builds on existing causal frameworks.
The paper tackles causal inference when confounders are functions of observed data, which violates positivity, and proposes two scenarios for estimable effects: functional interventions with a functional positivity condition, and nonparametric estimation using gradient fields. It introduces Level-set Orthogonal Descent Estimation (LODE), proves error bounds, and demonstrates results on simulated and real data.
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.