  The presence of unobserved common causes and the presence of measurement
error are two of the most limiting challenges in the task of causal structure
learning. Ignoring either of the two challenges can lead to detecting spurious
causal links among variables of interest. In this paper, we study the problem
of causal discovery in systems where these two challenges can be present
simultaneously. We consider linear models which include four types of
variables: variables that are directly observed, variables that are not
directly observed but are measured with error, the corresponding measurements,
and variables that are neither observed nor measured. We characterize the
extent of identifiability of such model under separability condition (i.e., the
matrix indicating the independent exogenous noise terms pertaining to the
observed variables is identifiable) together with two versions of faithfulness
assumptions and propose a notion of observational equivalence. We provide
graphical characterization of the models that are equivalent and present a
recovery algorithm that could return models equivalent to the ground truth.
