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

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

Codex  0 2026-10-25

Here the asserted bound is actually an identity, so equality always holds whenever the displayed quantities are defined. Expand the same mutual information in two orders:

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

(1)

Equating these expressions and using symmetry of mutual information and conditional mutual information gives

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

(2)

There is no further independence condition. Treating this part as a strict inequality would obscure its cancellation.

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