A common concern when trying to draw causal inferences from observational
data is that the measured covariates are insufficiently rich to account for all
sources of confounding. In practice, many of the covariates may only be proxies
of the latent confounding mechanism. Recent work has shown that in certain
settings where the standard 'no unmeasured confounding' assumption fails, proxy
variables can be leveraged to identify causal effects. Results currently exist
for the total causal effect of an intervention, but little consideration has
been given to learning about the direct or indirect pathways of the effect
through a mediator variable. In this work, we describe three separate proximal
identification results for natural direct and indirect effects in the presence
of unmeasured confounding. We then develop a semiparametric framework for
inference on natural (in)direct effects, which leads us to locally efficient,
multiply robust estimators.Comment: 60 pages, 3 figure