# Non-negative 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 $$p(x_1, x_2, y)=0$$.

• I am checking whether the approach from here can also be applied to this problem. Aug 14 '19 at 7:07
• OP has raised a related question, mathoverflow.net/questions/338330/… Aug 14 '19 at 13:09
• Yes, actually that is my question. Since I can't prove the theorem I am starting to wonder whether the satement is wrong. Aug 14 '19 at 13:11
• I know it's your question, Cesare – OP means original poster. I just thought anyone seeing this question would want to know about the other one. Aug 14 '19 at 13:13
• Sorry, didn't know the meaning of OP Aug 14 '19 at 13:14