Causal Inference by Minimizing the Dual Norm of Bias: Kernel Matching & Weighting Estimators for Causal Effects

We consider the problem of estimating causal effects from observational data and propose a novel framework for matching- and weighting-based causal estimators. The framework is based on expressing the bias of a causal estimator as an operator on the unknown conditional expectation function of outcomes and formulating the dual norm of the bias as the norm of this operator with respect to a function space that represents the potential structure for outcomes. We give the term worst-case bias minimizing (WCBM) to estimators that minimize this quantity for some function space and show that a great variety of existing causal estimators belong to this family, including one-to-one matching (with or without replacement), coarsened exact matching, and mean-matched sampling. We propose a range of new, kernel-based matching and weighting estimators that arise when one minimizes the dual norm of the bias with respect to a reproducing kernel Hilbert space. Depending on the case, these estimators can be solved either in closed form, using quadratic optimization, or using integer optimization. We show that estimators based on universal kernels are consistent for the causal effect. In numerical experiments, the new, kernel-based estimators outperform all standard causal estimators in estimation error, providing a successful balance between generality and efficiency.