A minimal set of shannon-type inequalities for functional dependence structures

Abstract

Abstract-The minimal set of Shannon-type inequalities (referred to as elemental inequalities), plays a central role in determining whether a given inequality is Shannon-type. Often, there arises a situation where one needs to check whether a given inequality is a constrained Shannon-type inequality. Another important application of elemental inequalities is to formulate and compute the Shannon outer bound for multi-source multisink network coding capacity. Under this formulation, it is the region of feasible source rates subject to the elemental inequalities and network coding constraints that is of interest. Hence it is of fundamental interest to identify the redundancies induced amongst elemental inequalities when given a set of functional dependence constraints. In this paper, we characterize a minimal set of Shannon-type inequalities when functional dependence constraints are present.

Entropy Inequalities for Lattices

“…To prove extensivity (18) of → assume that cl(X) ⊇ Y. Using extensivity (15) we also get cl(X) ⊇ X. Combining these two ineqalities gives cl(X) ⊇ X ⊎ Y as desired.…”

“…The relation between functional dependence and lattices has been studied [7,[10][11][12][13][14]. The relation between lattices and functional dependencies is closely related to minimal sets of Shannon-type Inequalities [15,16]. Relations between functional dependencies and Bayesian networks have also been described [8,17].…”

“…Assume that cl(X) ⊇ cl (Y). Using extensivity (15) we get cl(Y) ⊇ Y. Transitivity of ⊇ then gives cl(X) ⊇ Y. Assume cl(X) ⊇ Y.…”

Entropy Inequalities for Lattices

“…The relation between functional dependence and lattices has been studied [7,[9][10][11][12][13]. The relation between lattices and functional dependencies is closely related to minimal sets of Shannon-type inequalities [14,15]. Relations between functional dependencies and Bayesian networks have also been described [8,16].…”

Entropy Inequalities for Lattices

“…The relation between functional dependence and lattices has previously been studied [7,[9][10][11][12][13]. This is also closely related to minimal sets of Shannon-type Inequalities [14,15]. Relations between functional dependencies and Bayesian networks have also been described [8,16].…”