  Given only data generated by a standard confounding graph with unobserved
confounder, the Average Treatment Effect (ATE) is not identifiable. To estimate
the ATE, a practitioner must then either (a) collect deconfounded data;(b) run
a clinical trial; or (c) elucidate further properties of the causal graph that
might render the ATE identifiable. In this paper, we consider the benefit of
incorporating a large confounded observational dataset (confounder unobserved)
alongside a small deconfounded observational dataset (confounder revealed) when
estimating the ATE. Our theoretical results suggest that the inclusion of
confounded data can significantly reduce the quantity of deconfounded data
required to estimate the ATE to within a desired accuracy level. Moreover, in
some cases -- say, genetics -- we could imagine retrospectively selecting
samples to deconfound. We demonstrate that by actively selecting these samples
based upon the (already observed) treatment and outcome, we can reduce sample
complexity further. Our theoretical and empirical results establish that the
worst-case relative performance of our approach (vs. a natural benchmark) is
bounded while our best-case gains are unbounded. Finally, we demonstrate the
benefits of selective deconfounding using a large real-world dataset related to
genetic mutation in cancer.
