Entropy Region and Convolution

Abstract

Abstract. The entropy region is constructed from vectors of random variables by collecting Shannon entropies of all subvectors. Its shape is studied here by means of polymatroidal constructions, notably by convolution. The closure of the region is decomposed into the direct sum of tight and modular parts, reducing the study to the tight part. The relative interior of the reduction belongs to the entropy region. Behavior of the decomposition under selfadhesivity is clarified. Results are specialized to and completed for the region of four random variables. This and computer experiments help to visualize approximations of a symmetrized part of the entropy region. Four-atom conjecture on the minimization of Ingleton score is refuted.

Book Inequalities

“…Section III describes the book, a special iterated adhesive extension. Generalizing results from [9] and [12], we prove that book extensions always exist when the spine of the book has one Central European University, Budapest, Hungary e-mail: csirmaz@renyi.hu Research partially supported by TAMOP-4.2.2.C-11/1/KONV-2012-0001 and by the Lendulet program of the Hungarian Academy of Sciences element, or has all but one elements of the ground set. Sections IV and V concentrate on the case N = 4.…”

“…The next theorem is a generalization of [12,Theorem 3]. It will be used in proving the main result of this section, Theorem 6, and it essentially shows that for book extensions it is enough to consider tight polymatroids.…”

“…As g = g↓a if and only if g(N ) = g(N − {a}), it follows that g = g↓N if and only if every co-singleton has full rank. Such polymatroids are called tight in [12].…”

“…For example, B 4 contains 12 inequalities, eight of them come from the downward closed subsets depicted on Figure 1. The x s , y s , z s coefficients in the order above are (3,3,3), (4,5,3), (6,9,5), (7,12,5), (6,11,3), (4,7,1), (3,6,0), (5,8,3), and (5,8,3). The inequality coming from the last two triplets is a consequence of the others, as it is just the average of the inequalities coming from the triplets (4,5,3), and (6,11,3).…”

“…The method of Zhang and Yeung motivated the definition of adhesive extensions of polymatroids by F. Matúš in [9]. An alternative technique for generating non-Shannon inequalities was suggested by K. Makarychev et al [8], which later was found to rely on the same extension property of entropic polymatroids [5], [12]. Section II recalls some notation and terminology related to polymatroids; for a detailed account, see [7].…”