Sequential Kernel Embedding for Mediated and Time-Varying Dose Response Curves

TL;DR

This work develops sequential kernel embedding, a reproducing kernel Hilbert space (RKHS) method to estimate mediated and time-varying dose–response curves with continuous treatments and treatment–confounder feedback. By encoding conditional distributions and sequential plans into RKHS embeddings, the authors decouple the components of Pearl's mediation formula and Robins' g-formula, yielding simple, closed-form kernel ridge regression estimators with nonasymptotic uniform rates and, in discrete cases, semiparametric efficiency. The approach unifies mediation and dynamic treatment effect estimation under a flexible, nonparametric framework and is validated through nonlinear simulations and an empirical analysis of the US Job Corps data, with a cleaned data set provided as a benchmark. This method offers a practical, theoretically grounded path for policy evaluation where treatments and covariates are continuous and feedback loops are present, and it opens avenues for extending to counterfactual distributions and off-policy planning.

Abstract

We propose simple nonparametric estimators for mediated and time-varying dose response curves based on kernel ridge regression. By embedding Pearl's mediation formula and Robins' g-formula with kernels, we allow treatments, mediators, and covariates to be continuous in general spaces, and also allow for nonlinear treatment-confounder feedback. Our key innovation is a reproducing kernel Hilbert space technique called sequential kernel embedding, which we use to construct simple estimators that account for complex feedback. Our estimators preserve the generality of classic identification while also achieving nonasymptotic uniform rates. In nonlinear simulations with many covariates, we demonstrate strong performance. We estimate mediated and time-varying dose response curves of the US Job Corps, and clean data that may serve as a benchmark in future work. We extend our results to mediated and time-varying treatment effects and counterfactual distributions, verifying semiparametric efficiency and weak convergence.

Figures (7)

• Figure 1: Nonparametric response simulations. For the mediated response, we implement two estimators: huber2018direct (IPW, checkered gray) and our own (RKHS, white). For the time-varying dose response, we implement four estimators. From left to right, these are singh2020kernel {RKHS(ATE) , checkered white}, singh2020kernel {RKHS(CATE) , lined white}, lewis2020double (SNMM, gray), and our own {RKHS(GF) , white}.
• Figure 2: Effect of job training on arrests. We implement our estimators for total, direct, and indirect response curves (RKHS, solid).
• Figure 3: Effect of job training on employment. We implement our estimators for time-varying dose and incremental response curves {RKHS(GF) , solid}.
• Figure 4: Class hours for different samples.
• Figure 5: Effect of job training on arrests: $D\geq 40$. We implement our estimators for total, direct, and indirect response curves (RKHS, solid).

Theorems & Definitions (80)

• Remark 3.1: RKHS versus $\mathbb{L}^2$ inner product
• Definition 4.1: Pure mediated response curves robins1992identifiability
• Remark 4.1: Interventional mediated response curves
• Proposition 4.1: Convenient expressions
• Lemma 4.1: Pearl's mediation formula
• Remark 4.2: Pearl's mediation formula is unbounded over $\mathbb{L}^2$ when the treatment is continuous
• Remark 4.3: Mediational g-formula
• Theorem 4.1: Decoupling via sequential kernel embeddings
• proof : Proof sketch
• Remark 4.4: Key innovation