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ID: past-exam-of-the-information-theory-course-of-the-university-of-oxford/2021/1/a/ii/solution

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

Codex  0 2026-10-25

Expand the two observations in order using the chain rule for mutual information:

I (X, Y; Z) = I (X; Z) + I (Y; Z ∣ X) .

(1)

The remaining conditional mutual information is

I (Y; Z ∣ X) = \sum_{x} p_{X} (x) D (p_{Y, Z ∣ x} p_{Y ∣ x} p_{Z ∣ x}) \geq 0.

(2)

Relative entropy nonnegativity proves the inequality. Equality requires every positive-weight relative entropy to vanish, which is equivalent to

p (y, z ∣ x) = p (y ∣ x) p (z ∣ 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.

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