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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Conditional mutual information is concave in the channel input law

Statement

Let $Q, X, Z, Y$ be finite random variables and suppose the channel law $P_{Y ∣ X, Z, Q}$ depends on $Q$ only through $(X, Z)$ ; equivalently, $Q - (X, Z) - Y$ is a Markov chain. Then $I (X; Y ∣ Z) \geq I (X; Y ∣ Z, Q) .$ Thus, for a fixed finite channel and fixed side information $Z$ , averaging input laws over a time-sharing variable cannot increase the average conditional mutual information.

Facts & Assumptions

Given: The finite variables and Markov condition in the statement.

[F1]

Finite conditional mutual information: conditional mutual information is defined by the finite conditional-KL sum.

[F2]

Finite entropy and mutual-information chain rules: finite entropy and mutual information obey the chain rules used to expand the joint conditional mutual information.

[F3]

Finite conditional mutual information identities and equality: conditional mutual information is nonnegative.

Proof

1.1F1F2givenalgebra

By the chain rule in [F2], $I (X, Q; Y ∣ Z) = I (X; Y ∣ Z) + I (Q; Y ∣ X, Z) = I (X; Y ∣ Z),$ since the Markov assumption makes the last term zero.

2.1F2F3step 1.1givenalgebra∎

Expanding in the other order gives $I (X, Q; Y ∣ Z) = I (Q; Y ∣ Z) + I (X; Y ∣ Z, Q) .$ The first term on the right is nonnegative by [F3], proving the inequality.

Depends on

Used by

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Sources

Abbas El Gamal and Young-Han Kim, Lecture Notes on Network Information Theory, arXiv:1001.3404v4 (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Cambridge University Press prepublication book (standard reference, not scraped)