  Characteristic imsets are 0/1-vectors representing directed acyclic graphs
whose edges represent direct cause-effect relations between jointly distributed
random variables. A characteristic imset (CIM) polytope is the convex hull of a
collection of characteristic imsets. CIM polytopes arise as feasible regions of
a linear programming approach to the problem of causal disovery, which aims to
infer a cause-effect structure from data. Linear optimization methods typically
require a hyperplane representation of the feasible region, which has proven
difficult to compute for CIM polytopes despite continued efforts. We solve this
problem for CIM polytopes that are the convex hull of imsets associated to DAGs
whose underlying graph of adjacencies is a tree. Our methods use the theory of
toric fiber products as well as the novel notion of interventional CIM
polytopes. Our solution is obtained as a corollary of a more general result for
interventional CIM polytopes. The identified hyperplanes are applied to yield a
linear optimization-based causal discovery algorithm for learning polytree
causal networks from a combination of observational and interventional data.
