  Marginal polytopes are important geometric objects that arise in statistics
as the polytopes underlying hierarchical log-linear models. These polytopes can
be used to answer geometric questions about these models, such as determining
the existence of maximum likelihood estimates or the normality of the
associated semigroup. Cut polytopes of graphs have been useful in analyzing
binary marginal polytopes in the case where the simplicial complex underlying
the hierarchical model is a graph. We introduce a generalized cut polytope that
is isomorphic to the binary marginal polytope of an arbitrary simplicial
complex via a generalized covariance map. This polytope is full dimensional in
its ambient space and has a natural switching operation among its facets that
can be used to deduce symmetries between the facets of the correlation and
binary marginal polytopes. We find complete H-representations of the
generalized cut polytope for some important families of simplicial complexes.
We also compute the volume of these polytopes in some instances.
