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Marginal problems naturally arise in a variety of different fields:basically, the question is whether some marginal/partial information iscompatible with a joint probability distribution. To this aim, thecharacterization of marginal sets via quantifier elimination and polyhedralprojection algorithms is of primal importance. In this work, before consideringspecific problems, we review polyhedral projection algorithms with focus onapplications in information theory, and, alongside known algorithms, we alsopresent a newly developed geometric algorithm which walks along the facelattice of the polyhedron in the projection space. One important application ofthis is in the field of quantum non-locality, where marginal problems arise inthe computation of Bell inequalities. We apply the discussed algorithms todiscover many tight entropic Bell inequalities of the tripartite Bell scenarioas well as more complex networks arising in the field of causal inference.Finally, we analyze the usefulness of these inequalities as nonlocalitywitnesses by searching for violating quantum states.

Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling