Past exam of the information theory course of the University of Oxford / 2021 / 1 / a / ii / Solution

Expand the two observations in order using the chain rule for mutual information:The remaining conditional mutual information isRelative entropy nonnegativity proves the inequality. Equality requires every positive-weight relative entropy to vanish, which is equivalent to $p(y,z\mid x)=p(y\mid x)p(z\mid x)$ for every $x$ of positive probability. Thus equality holds exactly when $Y$ and $Z$ have conditional independence given $X$. No unconditional independence is required.