Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach

arXiv:2103.14029v5 Announce Type: replace Abstract: We study the estimation of causal parameters when not all confounders are observed and instead negative controls are available. Recent work has shown how these can enable identification and efficient estimation via two so-called bridge functions. In this paper, we tackle the primary challenges in causal inference with negative controls: identifying and estimating these bridge functions. Previous work has typically relied on completeness conditions for identification, imposed uniqueness assumptions for estimation, and focused on parametric models for the bridge functions. Instead, we provide an alternative identification strategy that assumes the existence of two bridge functions rather than completeness conditions. We also provide new estimators for these functions based on minimax learning formulations. These estimators accommodate general function classes such as Reproducing Kernel Hilbert Spaces and neural networks. We study finite-sample convergence results both for estimating bridge functions themselves and for the final estimation of the causal parameter under a variety of combinations of assumptions. We avoid uniqueness conditions on the bridge functions as much as possible.