  The abilities of humans to understand the world in terms of cause and effect
relationships, as well as to compress information into abstract concepts, are
two hallmark features of human intelligence. These two topics have been studied
in tandem in the literature under the rubric of causal abstractions theory. In
practice, it remains an open problem how to best leverage abstraction theory in
real-world causal inference tasks, where the true mechanisms are unknown and
only limited data is available. In this paper, we develop a new family of
causal abstractions by clustering variables and their domains. This approach
refines and generalizes previous notions of abstractions to better accommodate
individual causal distributions that are spawned by Pearl's causal hierarchy.
We show that such abstractions are learnable in practical settings through
Neural Causal Models (Xia et al., 2021), enabling the use of the deep learning
toolkit to solve various challenging causal inference tasks -- identification,
estimation, sampling -- at different levels of granularity. Finally, we
integrate these results with representation learning to create more flexible
abstractions, moving these results closer to practical applications. Our
experiments support the theory and illustrate how to scale causal inferences to
high-dimensional settings involving image data.
