Mutual information matrices are not always positive semi-definite
Abstract
For discrete random variables X1,..., Xn we construct an n by n matrix. In the (i,j) entry we put the mutual information I(Xi;Xj) between Xi and Xj. In particular, in the (i,i) entry we put the entropy H(Xi)=I(Xi;Xi) of Xi. This matrix, called the mutual information matrix of (X1,...,Xn), 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.
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