  We address the problem of causal effect estimation in the presence of
unobserved confounding, but where proxies for the latent confounder(s) are
observed. We propose two kernel-based methods for nonlinear causal effect
estimation in this setting: (a) a two-stage regression approach, and (b) a
maximum moment restriction approach. We focus on the proximal causal learning
setting, but our methods can be used to solve a wider class of inverse problems
characterised by a Fredholm integral equation. In particular, we provide a
unifying view of two-stage and moment restriction approaches for solving this
problem in a nonlinear setting. We provide consistency guarantees for each
algorithm, and we demonstrate these approaches achieve competitive results on
synthetic data and data simulating a real-world task. In particular, our
approach outperforms earlier methods that are not suited to leveraging proxy
variables.
