Maybe this is a very trivial question but my own answer to it is rather based on intuition only. Consider two random variables A and B. Their mutual information is I_AB. Now, I want to obtain information about B, but it is not possible. I can only access A such that my own variable C shares some information with A, I_AC. In this case, what would be the mutual Information between my variable C and the inaccessible variable B, that would be I_BC? My intuition would say that it must be I_BC = I_AC*I_AB / I_AA where I_AA is the full information of A. Is this intuition correct? Is there a proof of this, somewhere?
best regards
There is nothing you can really say about the mutual information $I(B;C)$ other than that it is smaller than $I(A;B)$ (this is the so-called data processing inequality). The problem to maximize $I(B;C)$ for a given $I(A;C)$ is dealt with in the information bottleneck problem.