  The need for modelling causal knowledge at different levels of granularity
arises in several settings. Causal Abstraction provides a framework for
formalizing this problem by relating two Structural Causal Models at different
levels of detail. Despite increasing interest in applying causal abstraction,
e.g. in the interpretability of large machine learning models, the graphical
and parametrical conditions under which a causal model can abstract another are
not known. Furthermore, learning causal abstractions from data is still an open
problem. In this work, we tackle both issues for linear causal models with
linear abstraction functions. First, we characterize how the low-level
coefficients and the abstraction function determine the high-level coefficients
and how the high-level model constrains the causal ordering of low-level
variables. Then, we apply our theoretical results to learn high-level and
low-level causal models and their abstraction function from observational data.
In particular, we introduce Abs-LiNGAM, a method that leverages the constraints
induced by the learned high-level model and the abstraction function to speedup
the recovery of the larger low-level model, under the assumption of
non-Gaussian noise terms. In simulated settings, we show the effectiveness of
learning causal abstractions from data and the potential of our method in
improving scalability of causal discovery.
