  Treatment effect estimation from observational data is a fundamental problem
in causal inference. There are two very different schools of thought that have
tackled this problem. On one hand, Pearlian framework commonly assumes
structural knowledge (provided by an expert) in form of directed acyclic graphs
and provides graphical criteria such as back-door criterion to identify valid
adjustment sets. On other hand, potential outcomes (PO) framework commonly
assumes that all observed features satisfy ignorability (i.e., no hidden
confounding), which in general is untestable. In prior works that attempted to
bridge these frameworks, there is an observational criteria to identify an
anchor variable and if a subset of covariates (not involving the anchor
variable) passes a suitable conditional independence criteria, then that subset
is a valid back-door. Our main result strengthens these prior results by
showing that under a different expert-driven structural knowledge -- that one
variable is a direct causal parent of treatment variable -- remarkably, testing
for subsets (not involving the known parent variable) that are valid back-doors
is equivalent to an invariance test. Importantly, we also cover the non-trivial
case where entire set of observed features is not ignorable (generalizing the
PO framework) without requiring knowledge of all parents of treatment variable.
Our key technical idea involves generation of a synthetic sub-sampling (or
environment) variable that is a function of the known parent variable. In
addition to designing an invariance test, this sub-sampling variable allows us
to leverage Invariant Risk Minimization, and thus, connects finding valid
adjustments (in non-ignorable observational setting) to representation
learning. We demonstrate effectiveness and tradeoffs of our approaches on a
variety of synthetic data as well as real causal effect estimation benchmarks.
