  Identification theory for causal effects in causal models associated with
hidden variable directed acyclic graphs (DAGs) is well studied. However, the
corresponding algorithms are underused due to the complexity of estimating the
identifying functionals they output. In this work, we bridge the gap between
identification and estimation of population-level causal effects involving a
single treatment and a single outcome. We derive influence function based
estimators that exhibit double robustness for the identified effects in a large
class of hidden variable DAGs where the treatment satisfies a simple graphical
criterion; this class includes models yielding the adjustment and front-door
functionals as special cases. We also provide necessary and sufficient
conditions under which the statistical model of a hidden variable DAG is
nonparametrically saturated and implies no equality constraints on the observed
data distribution. Further, we derive an important class of hidden variable
DAGs that imply observed data distributions observationally equivalent (up to
equality constraints) to fully observed DAGs. In these classes of DAGs, we
derive estimators that achieve the semiparametric efficiency bounds for the
target of interest where the treatment satisfies our graphical criterion.
Finally, we provide a sound and complete identification algorithm that directly
yields a weight based estimation strategy for any identifiable effect in hidden
variable causal models.
