Causal Estimation with Functional Confounders

TL;DR

This paper explores causal inference when confounders are functions of observed data, proposing new conditions and methods like LODE for effect estimation despite positivity violations.

Contribution

It introduces the concept of estimation with functional confounders, develops conditions for effect estimation, and proposes the LODE method with theoretical error bounds.

Findings

LODE provides effective effect estimates under functional confounder conditions

Conditions for functional positivity enable causal effect estimation

Empirical results validate the proposed methods on real and simulated data

Abstract

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.