  Scientists often pose questions about treatment effects on outcomes
conditional on a post-treatment event. However, defining, identifying, and
estimating causal effects conditional on post-treatment events requires care,
even in perfectly executed randomized experiments. Recently, the conditional
separable effect (CSE) was proposed as an interventionist estimand,
corresponding to scientifically meaningful questions in these settings.
However, while being a single-world estimand, which can be queried
experimentally, existing identification results for the CSE require no
unmeasured confounding between the outcome and post-treatment event. This
assumption can be violated in many applications. In this work, we address this
concern by developing new identification and estimation results for the CSE in
the presence of unmeasured confounding. We establish nonparametric
identification of the CSE in both observational and experimental settings when
certain proxy variables are available for hidden common causes of the
post-treatment event and outcome. We characterize the efficient influence
function for the CSE under a semiparametric model of the observed data law in
which nuisance functions are a priori unrestricted. Moreover, we develop a
consistent, asymptotically linear, and locally semiparametric efficient
estimator of the CSE using modern machine learning theory. We illustrate our
framework with simulation studies and a real-world cancer therapy trial.
