Partition-Symmetrical Entropy Functions

DOI10.1109/TIT.2016.2600580zbMATH Open1359.94306arXiv1407.7405OpenAlexW3102739603MaRDI QIDQ2976549

Author name not available (Why is that?)

Publication date: 28 April 2017

Published in: IEEE Transactions on Information Theory (Search for Journal in Brave)

Abstract: Let $calN=1,cdots,n$. The entropy function $$ of a set of $n$ discrete random variables $X_i:iincalN$ is a $2^n$-dimensional vector whose entries are $$, the (joint) entropies of the subsets of the set of $n$ random variables with $H(X_{e}mptyset)=0$ by convention. The set of all entropy functions for $n$ discrete random variables, denoted by $Gamm{a}_n^{*}$, is called the entropy function region for $n$. Characterization of $Gamm{a}_n^{*}$ and its closure $overlineGamm{a}_n^{*}$ 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 $calN$. An entropy function $$ is called $p$-symmetrical if for all $calA,calBsubsetcalN$, $$ whenever $|calAcap{calN}_i|=|calBcap{calN}_i|$, $i=1,cdots,t$. The set of all the $p$-symmetrical entropy functions, denoted by $Ps{i}_p^{*}$, is called $p$-symmetrical entropy function region. We prove that $overlinePs{i}_p^{*}$, the closure of $Ps{i}_p^{*}$, 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.

Full work available at URL: https://arxiv.org/abs/1407.7405

Mathematics Subject Classification ID

Measures of information, entropy (94A17)

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