Mutual information and Independence

Question

Let X, Y, Z be 3 random variables such that X and Z are independent. then can I say that I(X;Y|Z) = I(X;Y). and why?

Answer 1

Independence of $X$ and $Z$ is not enough to guarantee $I(X;Y|Z) = I(X;Y)$.

As an example consider $X \oplus Z = Y$, where $X,Z$ are independent Bernoulli($\frac12$) {0,1} random variables and $\oplus$ is modulo 2 addition.

$I(X;Y|Z) = H(X|Z) - H(X|YZ) = H(X) - H((Y \oplus Z)|YZ)= 1 - 0 = 1$.

On the other hand,

$I(X;Y) = H(X) - H(X|Y) = 1 - 1 = 0$.

Put in another way, $X$ and $Y$ are independent(prove?), and therefore there is no mutual information between them.