I am freely using known terms from communication complexity.
Let $u$ be a distribution on $X\times Y$, and $P$ a randomized (private randomness) communication protocol.
We define $IC(u,P) = I(P;X|Y)+I(P;Y|X)$ , where the probability space is both $u$, and the public randomness which are independent.
How do I show that $IC(u,P) \leq E[P_A] + E[P_B]$? Where $P_A$ is how many bits player $A$ sent, and $P_B$ how many bits player $B$ sent.
In fact more is true, $I(P;X|Y) \leq E[P_A]$, indeed intuitively player $B$ can learn from the protocol about $X$ at most what player $A$ says.
I am able to prove this for deterministic protocols. But the randomness makes it hard for me to formalize. By viewing public randomness as distribution on deterministic protocols, I can show the inequality if we have $I(P;X|Y,r)$ with $r$ the public randomness, and intuitively that should be enough, since player $B$ learns more from the protocol about $X$ if he knows what the protocol does. Still, I don't see how to prove this.
I'm actually not even sure if the definition purposely doesn't let player $A$ to know his private randomness. Trying to read braverman lecture notes but they're really hard to understand.