Partition-Symmetrical Entropy Functions

TL;DR

This paper characterizes the set of partition-symmetrical entropy functions, showing they are fully described by Shannon-type inequalities only in specific partition cases, which aids in solving symmetric information theory problems.

Contribution

It provides a complete characterization of the closure of partition-symmetrical entropy functions using Shannon-type inequalities for certain partitions, advancing understanding of entropy regions.

Findings

Closure of p-symmetrical entropy functions is characterized by Shannon inequalities for 1- and certain 2-partitions.

The result simplifies the analysis of entropy regions with symmetry.

It offers potential applications in symmetric information theory problems.

Abstract

Let $\cal{N}=\{1,\cdots,n\}$. The entropy function $\bf h$ of a set of $n$ discrete random variables $\{X_i:i\in\cal N\}$ is a $2^n$-dimensional vector whose entries are ${\bf{h}}({\cal{A}})\triangleq H(X_{\cal{A}}),\cal{A}\subset{\cal N} $, the (joint) entropies of the subsets of the set of $n$ random variables with $H(X_\emptyset)=0$ by convention. The set of all entropy functions for $n$ discrete random variables, denoted by $\Gamma^*_n$, is called the entropy function region for $n$. Characterization of $\Gamma^*_n$ and its closure $\overline{\Gamma^*_n}$ are well-known open problems in information theory. They are important not only because they play key roles in information theory problems but also they are related to other subjects in mathematics and physics. In this paper, we consider \emph{partition-symmetrical entropy functions}. Let $p=\{\cal{N}_1,\cdots, \cal{N}_t\}$ be a $t$-partition of $\cal N$. An entropy function $\bf h$ is called $p$-symmetrical if for all ${\cal A},{\cal B} \subset {\cal N}$, $\bf{h}({\cal A}) = \bf{h}({\cal B})$ whenever $|{\cal A} \cap {\cal N}_i| = |{\cal B} \cap {\cal N}_i|$, $i = 1, \cdots,t$. The set of all the $p$-symmetrical entropy functions, denoted by $\Psi^*_p$, is called $p$-symmetrical entropy function region. We prove that $\overline{\Psi^*_p}$, the closure of $\Psi^*_p$, is completely characterized by Shannon-type information inequalities if and only if $p$ is the $1$-partition or a $2$-partition with one of its blocks being a singleton. The characterization of the partition-symmetrical entropy functions can be useful for solving some information theory and related problems where symmetry exists in the structure of the problems. Keywords: entropy, entropy function, information inequality, polymatroid.