= Solution
Expand the two observations in order using the <chain rule for mutual information>:
$$
I(X,Y;Z)=I(X;Z)+I(Y;Z\mid X).
$$
The remaining <conditional mutual information> is
$$
I(Y;Z\mid X)=\sum_xp_X(x)D\!\left(p_{Y,Z\mid x}\,\middle\|\,p_{Y\mid x}p_{Z\mid x}\right)\geq0.
$$
<Relative 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 \b[equality holds exactly when $Y$ and $Z$ have <conditional independence> given $X$.] No unconditional <independence> is required.
Back to article page
