Is it true that:

If I(X;Y) = 0 then I(S;X) + I(S;Y) <= I(S;X,Y)

Can you find a counterexample, or prove this and teach me your proof?

Someone showed me a simple analytic proof. I am still interested in seeing different ways people might prove this though.

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For a visualization, see information diagrams, and note that the central cell I(S; X; Y) must be non-positive (because I(S; X; Y) + I(X; Y | S) = I(X; Y) = 0).

We want to prove:

This can be rewritten as:

After moving everything to the right hand side and simplifying, we get:

Now if we just prove that is a probability distribution, then the left hand side is , and Kullback-Leibler divergence is always nonnegative.

Ok, q is obviously nonnegative, and its integral equals 1:


1 comment, sorted by Click to highlight new comments since: Today at 12:56 PM

Just for amusement, I think this theorem can fail when s, x, y represent subsystems of an entangled quantum state. (The most natural generalization of mutual information to this domain is sometimes negative.)