A Conditional Information Inequality and Its Combinatorial Applications

We show that the inequality H(A|B, X) + H(A|B, Y) ≤ H(A|B) for jointly distributed random variables A, B, X, Y, which does not hold in general case, holds under some natural condition on the support of the probability distribution of A, B, X, Y. This result generalizes a version of the conditional Ingleton inequality: if for some distribution I(X : Y|A) = H(A|X, Y) = 0, then I(A : B) ≤ I (A : B|X)...