Approximate Causal Abstraction
  Scientific models describe natural phenomena at different levels of
abstraction. Abstract descriptions can provide the basis for interventions on
the system and explanation of observed phenomena at a level of granularity that
is coarser than the most fundamental account of the system. Beckers and Halpern
(2019), building on work of Rubenstein et al. (2017), developed an account of
abstraction for causal models that is exact. Here we extend this account to the
more realistic case where an abstract causal model offers only an approximation
of the underlying system. We show how the resulting account handles the
discrepancy that can arise between low- and high-level causal models of the
same system, and in the process provide an account of how one causal model
approximates another, a topic of independent interest. Finally, we extend the
account of approximate abstractions to probabilistic causal models, indicating
how and where uncertainty can enter into an approximate abstraction.
