TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Data processing for a finite Markov chain, with equality condition

Statement

Let $X, Y, Z$ have a joint pmf on finite alphabets and suppose that, for every $y$ with $p_{Y} (y) > 0$ ,

$p_{X Z ∣ y} (x, z) = p_{X ∣ y} (x) p_{Z ∣ y} (z) .$

This is the finite Markov condition $X \to Y \to Z$ . Then

$I (X; Z) \leq I (X; Y),$

with equality if and only if $I (X; Y ∣ Z) = 0$ , equivalently $X$ and $Y$ are conditionally independent given every positive-mass value of $Z$ .

Facts & Assumptions

Given: A finite joint pmf $p_{X Y Z}$ satisfying the stated Markov factorization.

[F1]

Finite entropy and mutual-information chain rules: $I (X; Y, Z) = I (X; Y) + I (X; Z ∣ Y) = I (X; Z) + I (X; Y ∣ Z)$ .

[F2]

Finite conditional mutual information identities and equality: finite CMI is nonnegative and vanishes exactly at conditional independence on each positive-mass conditioning value.

Proof

1.1givenF2algebra

For each $y$ with $p_{Y} (y) > 0$ , the hypothesis makes $p_{X Z ∣ y} = p_{X ∣ y} \otimes p_{Z ∣ y}$ . The corresponding conditional KL term in the definition of $I (X; Z ∣ Y)$ is therefore $D (p_{X Z ∣ y} ∥ p_{X ∣ y} \otimes p_{Z ∣ y}) = 0$ , so the finite weighted sum gives $I (X; Z ∣ Y) = 0$ .

2.1F1F2step 1.1algebra∎

Equating the two expansions in [F1] and using step 1.1 gives $I (X; Y) - I (X; Z) = I (X; Y ∣ Z)$ . This is nonnegative by [F2], proving data processing. The difference vanishes exactly when $I (X; Y ∣ Z) = 0$ , and [F2] identifies that equality case with conditional independence of $X, Y$ given each positive-mass $z$ .

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Robert M. Gray, Entropy and Information Theory, corrected first edition (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author prepublication (standard reference, not scraped)