Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam Region

We initiate the study of the Stam region, defined as the subset of the positive orthant in ℝ(2n-1) that arises from considering the entropy powers of subset sums of n independent random vectors in a Euclidean space of finite dimension. We show that the class of fractionally superadditive set functions provides an outer bound to the Stam region, resolving a conjecture of Barron and Madiman. On the other hand, the entropy power of a sum of independent random vectors is not supermodular in any dimension. We also develop some qualitative properties of the Stam region, showing for instance that its closure is a logarithmically convex cone.