RemarkRemark: Literature-sourcedProof: 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 and differential entropy are different quantities

Remark

For a probability law with Lebesgue density $f$ on $R^{d}$ , its differential entropy, when the integral is defined, is $h (f) : = - \int_{R^{d}} f (x) \ln f (x) d x .$ This density-based quantity is not the law entropy $H (X) = I (X; X)$ of Law entropy as self-mutual information. Differential entropy may be finite or negative even when law entropy is infinite; the uniform law on $[0, 1]$ is a basic example. No identity between the two is used in this page.

Depends on

Law entropy as self-mutual information

Used by

The uniform law has infinite self-information entropy and zero differential entropy Example

Dependency tree · two levels

3 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

John Duchi, Lecture Notes on Statistics and Information Theory, Chapter 2 §2.1 (standard reference, not scraped)
Robert M. Gray, Entropy and Information Theory, corrected author-hosted first edition, Chapter 1 and Chapter 5 §5.5 (standard reference, not scraped)