  We focus on causal discovery in the presence of measurement error in linear
systems where the mixing matrix, i.e., the matrix indicating the independent
exogenous noise terms pertaining to the observed variables, is identified up to
permutation and scaling of the columns. We demonstrate a somewhat surprising
connection between this problem and causal discovery in the presence of
unobserved parentless causes, in the sense that there is a mapping, given by
the mixing matrix, between the underlying models to be inferred in these
problems. Consequently, any identifiability result based on the mixing matrix
for one model translates to an identifiability result for the other model. We
characterize to what extent the causal models can be identified under a
two-part faithfulness assumption. Under only the first part of the assumption
(corresponding to the conventional definition of faithfulness), the structure
can be learned up to the causal ordering among an ordered grouping of the
variables but not all the edges across the groups can be identified. We further
show that if both parts of the faithfulness assumption are imposed, the
structure can be learned up to a more refined ordered grouping. As a result of
this refinement, for the latent variable model with unobserved parentless
causes, the structure can be identified. Based on our theoretical results, we
propose causal structure learning methods for both models, and evaluate their
performance on synthetic data.
