  Missing data has the potential to affect analyses conducted in all fields of
scientific study, including healthcare, economics, and the social sciences.
Several approaches to unbiased inference in the presence of non-ignorable
missingness rely on the specification of the target distribution and its
missingness process as a probability distribution that factorizes with respect
to a directed acyclic graph. In this paper, we address the longstanding
question of the characterization of models that are identifiable within this
class of missing data distributions. We provide the first completeness result
in this field of study -- necessary and sufficient graphical conditions under
which, the full data distribution can be recovered from the observed data
distribution. We then simultaneously address issues that may arise due to the
presence of both missing data and unmeasured confounding, by extending these
graphical conditions and proofs of completeness, to settings where some
variables are not just missing, but completely unobserved.
