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We address the problem of causal effect estimation in the presence ofunobserved confounding, but where proxies for the latent confounder(s) areobserved. We propose two kernel-based methods for nonlinear causal effectestimation in this setting: (a) a two-stage regression approach, and (b) amaximum moment restriction approach. We focus on the proximal causal learningsetting, but our methods can be used to solve a wider class of inverse problemscharacterised by a Fredholm integral equation. In particular, we provide aunifying view of two-stage and moment restriction approaches for solving thisproblem in a nonlinear setting. We provide consistency guarantees for eachalgorithm, and we demonstrate these approaches achieve competitive results onsynthetic data and data simulating a real-world task. In particular, ourapproach outperforms earlier methods that are not suited to leveraging proxyvariables.

Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction