  Causal effect identification considers whether an interventional probability
distribution can be uniquely determined without parametric assumptions from
measured source distributions and structural knowledge on the generating
system. While complete graphical criteria and procedures exist for many
identification problems, there are still challenging but important extensions
that have not been considered in the literature. To tackle these new settings,
we present a search algorithm directly over the rules of do-calculus. Due to
generality of do-calculus, the search is capable of taking more advanced
data-generating mechanisms into account along with an arbitrary type of both
observational and experimental source distributions. The search is enhanced via
a heuristic and search space reduction techniques. The approach, called
do-search, is provably sound, and it is complete with respect to
identifiability problems that have been shown to be completely characterized by
do-calculus. When extended with additional rules, the search is capable of
handling missing data problems as well. With the versatile search, we are able
to approach new problems such as combined transportability and selection bias,
or multiple sources of selection bias. We perform a systematic analysis of
bivariate missing data problems and study causal inference under case-control
design. We also present the R package dosearch that provides an interface for a
C++ implementation of the search.
