# Non-negativity of interaction information for special trivariate case

Consider a discrete trivariate distribution $$P(X_1, X_2, Y)$$, which satisfies $$p(x_1, x_2, y) = \min( p(x_1,y), p(x_2,y) ),$$ for all $$x_1$$ and $$x_2$$ for which $$p(x_1, x_2) > 0$$ and for all values of $$y$$ (including those at which $$p(x_1, x_2, y) = 0$$). I am trying to show that then the interaction-information is non-negative $$I(X_1;Y) + I(X_2;Y) - I(X_1 X_2;Y) \ge 0$$

I need either a rigorous proof or, alternatively, a counterexample that disproves the theorem.

Addendum 1: I use the shorthand notation $$p(x_1, x_2, y) := P(X_1=x_1, X_2 = x_2, Y=y)$$

Addendum 2: for a similar inequality and its proof see here. This might provide some inspiration. For theoretical references on non-Shannon inequalities see here and here.

Addendum 3: Note that the canonical example of negative interaction-information, the XOR gate, does not satisfy the theorem hypothesis. For the XOR gate $$p(x_1, x_2, y) = \min \left( \ p(x_1,y), \ p(x_2,y) \ \right)$$ holds for all $$x_1, x_2, y$$ such that $$p(x_1, x_2, y)>0$$ but not for all $$y$$ once we pick $$x_1, x_2$$ points for which $$p(x_1, x_2)>0$$, i.e. also the $$y$$s for which $$p(x_1, x_2, y)=0$$.

• Could you supply an example of such a distribution? Without an example, it looks possible that any answers could be discussing a situation that never exists and therefore will be devoid of information. Is it possible "$p$" refers to a cumulative distribution function? – whuber Jun 23 '19 at 12:13
• Actually, it is pretty common. Below an example, which I have built so that it does not look excessively trivial. $P$ is a discrete distribution. I have corrected the notation to make it more clear. Example: $$\begin{array}{c|c|c|C} X_1 & X_2 & Y & P\\ 1 & 2 & 1 & 1/5 \\ 2 & 2 & 1 & 1/5 \\ 2 & 2 & 2 & 1/5 \\ 2 & 1 & 3 & 1/5 \\ 3 & 3 & 4 & 1/5 \\ \end{array}$$ – Cesare Jun 23 '19 at 19:06
• That's a trivial example, because conditional on $Y,$ almost surely at least one variable is constant. Can you offer any example of substance? – whuber Jun 24 '19 at 12:42
• Hint: what's the information $I(X)$ for a random variable $X$ that is almost surely a constant? – whuber Jun 24 '19 at 13:58
• You cannot compute information for one variable. – Cesare Jun 24 '19 at 18:05