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Past exam of the communication Theory course of the University of Oxford / 2010 / 3 / b / Solution by

Codex  0 2026-10-31

Fix the channel matrix

W

. The mixed input induces the mixed output

q_{3} = λ q_{1} + (1 - λ) q_{2}

. The concavity of information entropy implies

H (q_{3}) \geq λ H (q_{1}) + (1 - λ) H (q_{2}) .

For completeness, the difference is

λ D (q_{1} ∥ q_{3}) + (1 - λ) D (q_{2} ∥ q_{3}) \geq 0

by Gibbs inequality, including zero coordinates with the usual conventions.

On the other hand, conditional output entropy is linear in the input law:

H (Y_{3} ∣ X_{3}) = \sum_{x} p_{3} (x) H (W (\cdot ∣ x)) = λ H (Y_{1} ∣ X_{1}) + (1 - λ) H (Y_{2} ∣ X_{2}) .

Subtracting gives the concavity of mutual information in the input distribution:

I (X_{3}; Y_{3}) \geq λ I (X_{1}; Y_{1}) + (1 - λ) I (X_{2}; Y_{2}) .

This makes maximizing mutual information over the input probability simplex a concave optimization problem: a local maximum is global. It also justifies averaging input distributions over channel symmetries when searching for an optimum, since symmetry averaging cannot lower the achieved mutual information. The channel itself is held fixed throughout.

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