Mutual Information Matrices Are Not Always Positive Semidefinite

For discrete random variables X₁, ..., 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 (X₁, ..., X_n), has been conjectured to be positive semidefinite. In this paper, we give counterexamples to the conjecture, and show that the conjecture holds for up to three random variables.