Causal Inference with Unmeasured Confounding: A Minimax Perspective

Abstract

Unmeasured confounding presents a fundamental challenge to causal inference in observational studies. Standard methods often rely on the untestable assumption that all common causes of treatment and outcome have been measured. This paper explores a minimax perspective for navigating this challenge. We frame the problem of causal estimation as a game against nature, where nature chooses the magnitude and direction of confounding from a plausible set of possibilities, and the statistician chooses an estimator to minimize the worst-case error. This approach leads to estimators that are robust to a specified degree of confounding. We formalize the classes of confounding scenarios and derive the corresponding minimax optimal estimators for the average treatment effect. The methodology provides not only a point estimate but also an interval that accounts for the uncertainty due to unmeasured confounding, offering a more honest assessment of causal claims. We demonstrate that the minimax estimator balances the trade-off between bias introduced by confounding and variance from estimation, providing a principled way to perform causal inference under ambiguity.