  Marginal problems naturally arise in a variety of different fields:
basically, the question is whether some marginal/partial information is
compatible with a joint probability distribution. To this aim, the
characterization of marginal sets via quantifier elimination and polyhedral
projection algorithms is of primal importance. In this work, before considering
specific problems, we review polyhedral projection algorithms with focus on
applications in information theory, and, alongside known algorithms, we also
present a newly developed geometric algorithm which walks along the face
lattice of the polyhedron in the projection space. One important application of
this is in the field of quantum non-locality, where marginal problems arise in
the computation of Bell inequalities. We apply the discussed algorithms to
discover many tight entropic Bell inequalities of the tripartite Bell scenario
as well as more complex networks arising in the field of causal inference.
Finally, we analyze the usefulness of these inequalities as nonlocality
witnesses by searching for violating quantum states.
