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

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

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Finite information quantities convert by the logarithm base

Statement

For any $b > 1$ , define each base- $b$ finite information quantity by replacing $\ln$ with $\log_{b}$ in its defining finite formula. Then

$H_{b} (p) = \frac{H (p)}{\ln b}, D_{b} (p ∥ q) = \frac{D (p ∥ q)}{\ln b},$

where the second equality is in $[0, + \infty]$ and $+ \infty / \ln b : = + \infty$ . The same conversion holds for finite conditional entropy, mutual information, and conditional mutual information. In particular, values in bits equal nats divided by $\ln 2$ .

Facts & Assumptions

Given: A base $b > 1$ and finite probability vectors or finite joint probability laws.

[F1]

Information units and logarithm-base convention: $\log_{b} t = \ln (t) / \ln (b)$ for $t > 0$ , $\ln b > 0$ , and zero-mass terms use $0 \log_{b} 0 = 0$ .

[F2]

Finite distributions, entropy, divergence, and stochastic matrices: finite entropy and KL use finite sums; a support failure gives KL value $+ \infty$ .

[F3]

Finite conditional entropy: $H (X ∣ Y)$ is a finite weighted sum of entropies of positive-mass conditional pmfs.

[F4]

Finite mutual information: $I (X; Y)$ is a finite sum over positive joint support.

[F5]

Finite conditional mutual information: $I (X; Y ∣ Z)$ is a finite weighted sum of conditional KL values over positive-mass $z$ .

Proof

1.1givenF1F2algebra

By [F1], $\ln b > 0$ , so division by $\ln b$ preserves order and sends $+ \infty$ to $+ \infty$ . For entropy, replacing each $\ln p_{i}$ by $\log_{b} p_{i} = \ln p_{i} / \ln b$ factors the same positive scalar out of the finite sum; zero terms remain zero by [F1]. The identical termwise calculation applies to every finite KL sum when $p$ is supported by $q$ .

2.1F1F3F4F5step 1.1algebra∎

If $p_{i} > 0 = q_{i}$ , both base conventions set the KL value to $+ \infty$ , agreeing with the extended-real equation. Conditional entropy is a finite weighted sum of the entropy terms from step 1.1 by [F3]; mutual information is a finite KL sum by [F4]; conditional mutual information is a finite weighted sum of conditional KL sums by [F5]. Factoring $1 / \ln b$ from each of these finite sums proves all remaining conversions.

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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)