Mutual Information Matrices Are Not Always Positive Semidefinite

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 semidefinite. In this paper, we give counterexamples to the conjecture, and show that the conjecture holds for up to three random variables.