# Mutual information equals conditional mutual information

Consider three random variables $X,Y,Z$. It is standard that $I(X,Y|Z)=0$ if and only if $X,Y$ are conditionally independent given $Z$. If instead we require $I(X,Y|Z)=I(X,Y)$, what do we get? What properties would have the probability distributions that satisfy such constraint?

You are in luck: this is a well-studied quantity under a different name. This is equivalent to the Interaction information or a measure of synergy being equal to zero. Consider the quantity $S(X,Y,Z) = I(X,Y|Z) - I(X,Y)$. In 2 this is identical to eq. 10 if you first re-write $I(X,Y|Z) = I(X, (Y,Z)) -I(X,Z)$, so that $S(X,Y,Z) = I(X, (Y,Z)) -I(X,Z) - I(X,Y)$. This quantity can be positive, negative or zero. If it is zero, it means that the information that $Y$ and $Z$ have about $X$ is additive. If it is negative it means that the information that $Y$ and $Z$ have about $X$ is redundant, and if it is positive the information is synergistic (i.e., Y and Z together have information about X that can not be gleaned from either one individually).
• In what they call the FM case, $I=-1$ but the probability distribution has only pairwise functions, so not every pairwise potential has zero interaction information. But does the viceversa hold, i.e. does $I=0$ imply only (up to) pairwise functions? Commented Sep 15, 2015 at 15:54