Mutual information matrices are not always positive semi-definite
Information Theory
2013-07-26 v1 math.IT
Abstract
For discrete random variables X_1,..., X_n we construct an n by n matrix. In the (i,j) entry we put the mutual information I(X_i;X_j) between X_i and X_j. In particular, in the (i,i) entry we put the entropy H(X_i)=I(X_i;X_i) of X_i. This matrix, called the mutual information matrix of (X_1,...,X_n), has been conjectured to be positive semi-definite. In this note, we give counterexamples to the conjecture, and show that the conjecture holds for up to three random variables.
Keywords
Cite
@article{arxiv.1307.6673,
title = {Mutual information matrices are not always positive semi-definite},
author = {Sune K. Jakobsen},
journal= {arXiv preprint arXiv:1307.6673},
year = {2013}
}
Comments
4 pages, 1 figure