  An important task in data analysis is the discovery of causal relationships
between observed variables. For continuous-valued data, linear acyclic causal
models are commonly used to model the data-generating process, and the
inference of such models is a well-studied problem. However, existing methods
have significant limitations. Methods based on conditional independencies
(Spirtes et al. 1993; Pearl 2000) cannot distinguish between
independence-equivalent models, whereas approaches purely based on Independent
Component Analysis (Shimizu et al. 2006) are inapplicable to data which is
partially Gaussian. In this paper, we generalize and combine the two
approaches, to yield a method able to learn the model structure in many cases
for which the previous methods provide answers that are either incorrect or are
not as informative as possible. We give exact graphical conditions for when two
distinct models represent the same family of distributions, and empirically
demonstrate the power of our method through thorough simulations.
