DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

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.

Law entropy as self-mutual information

Definition

For a random element $X$ with law $P_{X}$ , assume AC on the absolutely continuous branch $P_{X X} ≪ P_{X} \otimes P_{X}$ , as in Measure-theoretic mutual information; the other branch has value $+ \infty$ without that premise. Define its law entropy by $H (X) : = I (X; X) = D (P_{X X} ∥ P_{X} \otimes P_{X}),$ where $P_{X X}$ is the joint law of the diagonal pair $(X, X)$ . This is an extended nonnegative quantity in nats and depends only on the law of X.

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 author-hosted first edition, Chapter 5 §5.5 (standard reference, not scraped)
John Duchi, Lecture Notes on Statistics and Information Theory, Chapter 2 §2.1 (standard reference, not scraped)