  Proximal causal inference was recently proposed as a framework to identify
causal effects from observational data in the presence of hidden confounders
for which proxies are available. In this paper, we extend the proximal causal
inference approach to settings where identification of causal effects hinges
upon a set of mediators which are not observed, yet error prone proxies of the
hidden mediators are measured. Specifically, (i) We establish causal hidden
mediation analysis, which extends classical causal mediation analysis methods
for identifying natural direct and indirect effects under no unmeasured
confounding to a setting where the mediator of interest is hidden, but proxies
of it are available. (ii) We establish hidden front-door criterion, which
extends the classical front-door criterion to allow for hidden mediators for
which proxies are available. (iii) We show that the identification of a certain
causal effect called population intervention indirect effect remains possible
with hidden mediators in settings where challenges in (i) and (ii) might
co-exist. We view (i)-(iii) as important steps towards the practical
application of front-door criteria and mediation analysis as mediators are
almost always measured with error and thus, the most one can hope for in
practice is that the measurements are at best proxies of mediating mechanisms.
We propose identification approaches for the parameters of interest in our
considered models. For the estimation aspect, we propose an influence
function-based estimation method and provide an analysis for the robustness of
the estimators.
