Kernel Methods for Multistage Causal Inference: Mediation Analysis and Dynamic Treatment Effects

We propose simple estimators for mediation analysis and dynamic treatment effects over short horizons, which preserve the nonlinearity, dependence, and effect modification of identification theory. We allow treatments, mediators, and covariates to be discrete or continuous in general spaces. Across this broad variety of data settings, the estimators have closed form solutions in terms of kernel matrix operations due to our algorithmic innovation: sequential mean embedding of the mediator and covariate conditional distributions given a hypothetical treatment sequence. The simple estimators have strong guarantees. For the continuous treatment case, we prove uniform consistency with finite sample rates that match the minimax optimal rate for standard kernel ridge regression. For the discrete treatment case, we prove $n^{-1/2}$ consistency, finite sample Gaussian approximation, and semiparametric efficiency. We extend the analysis to incremental effects and counterfactual distributions, identifying and estimating new causal estimands. In nonlinear simulations with many covariates, we demonstrate state-of-the-art performance. We estimate mediated and dynamic treatment effects of the US Job Corps program for disadvantaged youth, and share a cleaned data set that may serve as a benchmark in future work.