DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite mutual information

Definition

Let $p_{X Y}$ be a joint probability vector on finite alphabets $X \times Y$ , with marginals $p_{X} (x) : = \sum_{y} p_{X Y} (x, y)$ and $p_{Y} (y) : = \sum_{x} p_{X Y} (x, y)$ . The marginals are probability vectors by Marginals and positive-mass conditionals of a finite joint law are probability vectors. The joint vector is the outcome-weight function of the canonical finite probability space from Finite probability spaces, outcome weights, events, and event probabilities; its coordinate maps are finite random elements in the sense of Random elements and real random variables. Define mutual information in nats by

$I (X; Y) : = D (p_{X Y} ∥ p_{X} \otimes p_{Y}) = \sum_{(x, y) : p_{X Y} (x, y) > 0} p_{X Y} (x, y) \ln \frac{p_{X Y} (x, y)}{p_{X} (x) p_{Y} (y)},$

where $D$ is the finite KL convention of Finite distributions, entropy, divergence, and stochastic matrices and $(p_{X} \otimes p_{Y}) (x, y) : = p_{X} (x) p_{Y} (y)$ . Every ratio in the displayed sum is positive and finite: if $p_{X Y} (x, y) > 0$ , then each marginal sum is at least that positive entry. The sum is over finite support, so $I (X; Y)$ is finite. The base- $b$ version replaces $\ln$ by $\log_{b}$ .

Depends on

Used by

…and 24 more results.

Dependency tree · two levels

17 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)