Compact representation of polymatroid axioms for random variables with conditional independencies

Abstract

The polymatroid axioms are dominantly used to study the capacity limits of various communication systems. In fact for most of the communication systems, for which the capacity is known, these axioms are solely required to obtain the characterization of capacity. Moreover, the polymatroid axioms are stronger tools to tackle the implication problem for conditional independencies compared to the axioms used in Bayesian networks. However, their use is prohibitively complex as the number of random variables increases since the number of inequalities to consider increases exponentially. In this paper we give a compact characterization of the minimal set of polymatroid axioms when arbitrary conditional independence and functional dependence constraints are given. In particular, we identify those elemental equalities which are implied by given constraints. We also identify those elemental inequalities which are redundant given the constraints.

Indistinguishability of causal relations from limited marginals

“…For instance, it was shown in Ref. [72] that if I(A : B|C) = 0, with A, B, C disjoint sets of variables, then from the set of polymatroid axioms in Eq. (11), the following inequalities are redundant…”

“…where E ∈ [n] \ {A, B} ∩ C. Hence, in this case one can generate a compact representation of polymatroid axioms [72].…”