A Group Theoretic Model for Information

TL;DR

This paper formalizes the concept of information elements using group theory, establishing a deep connection between information lattices and subgroup lattices, and explores implications for information laws and properties.

Contribution

It introduces a formal group-theoretic model for information elements, revealing isomorphisms and approximation relations between information and subgroup structures.

Findings

Isomorphisms between information lattices and subgroup lattices

Approximation of entropy structures by log-indices of subgroups

Counterexamples showing non-submodularity of common information

Abstract

In this paper we formalize the notions of information elements and information lattices, first proposed by Shannon. Exploiting this formalization, we identify a comprehensive parallelism between information lattices and subgroup lattices. Qualitatively, we demonstrate isomorphisms between information lattices and subgroup lattices. Quantitatively, we establish a decisive approximation relation between the entropy structures of information lattices and the log-index structures of the corresponding subgroup lattices. This approximation extends the approximation for joint entropies carried out previously by Chan and Yeung. As a consequence of our approximation result, we show that any continuous law holds in general for the entropies of information elements if and only if the same law holds in general for the log-indices of subgroups. As an application, by constructing subgroup counterexamples we find surprisingly that common information, unlike joint information, obeys neither the submodularity nor the supermodularity law. We emphasize that the notion of information elements is conceptually significant--formalizing it helps to reveal the deep connection between information theory and group theory. The parallelism established in this paper admits an appealing group-action explanation and provides useful insights into the intrinsic structure among information elements from a group-theoretic perspective.