Bounding the entropic region via information geometry

This paper suggests that information geometry may form a natural framework to deal with the unknown part of the boundary of entropic region. An application of information geometry shows that distributions associated with Shannon facets can be associated, in the right coordinates, with affine collections of distributions. This observation allows an information geometric reinterpretation of the Shannon-type inequalities as arising from a Pythagorean style relationship. The set of distributions which violate Ingleton´s inequality, and hence are linked with the part of the entropic region which is yet undetermined, is shown also to have a surprising affine information geometric structure in a special case involving four random variables and a certain support. These facts provide strong evidence for the link between information geometry and characterizing the boundary of the entropic region.