LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Finite relative entropy is continuous on a simplex with positive reference law

Statement

Let $A$ be finite and let $p$ be a probability vector with $p (a) > 0$ for every $a \in A$ . Then $q \mapsto D (q ∥ p)$ is finite and continuous on the probability simplex $Δ (A)$ .

Facts & Assumptions

Given: A finite alphabet $A$ , a full-support probability vector $p$ on $A$ , and the finite relative entropy convention.

[F1]

Finite distributions, entropy, divergence, and stochastic matrices: $D (q ∥ p) = \sum_{a : q (a) > 0} q (a) \ln (q (a) / p (a))$ , and $0 \ln 0 = 0$ ; if $q (a) > 0 = p (a)$ for some coordinate then $D (q ∥ p) = + \infty$ .

[F2]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: for $u > 0$ , $\ln u = \int_{1}^{u} d s / s$ .

Proof

1.1F1F2algebra

For $0 < t \leq 1$ , put $u = 1 / t \geq 1$ . Then $∣ t \ln t ∣ = (\ln u) / u$ , and by [F2], $0 \leq (\ln u) / u = (1 / u) \int_{1}^{u} d s / s \leq (1 / u) \int_{1}^{u} d s / \sqrt{s} \leq 2 / \sqrt{u} \to 0$ . Thus $g (t) = t \ln t$ extends continuously to $t = 0$ by $g (0) = 0$ .

2.1F1F2step 1.1algebra∎

Since every $p (a) > 0$ , the number $\ln p (a)$ is finite, and $D (q ∥ p) = \sum_{a \in A} g (q (a)) - \sum_{a \in A} q (a) \ln p (a)$ . Each summand is continuous in $q (a)$ by step 1.1; the second sum is linear and has finitely many terms. Hence $D (q ∥ p)$ is finite and continuous on the whole simplex, including its boundary.

Depends on

Used by

Dependency tree · two levels

10 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

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, second edition, Chapter 11 (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author-hosted book draft (standard reference, not scraped)