DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedprecheck pending

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 directed information and causal conditioning

Definition

For finite random sequences $X^{n} = (X_{1}, \dots, X_{n})$ and $Y^{n} = (Y_{1}, \dots, Y_{n})$ , define the directed information from $X^{n}$ to $Y^{n}$ by $I (X^{n} \to Y^{n}) : = \sum_{i = 1}^{n} I (X^{i}; Y_{i} ∣ Y^{i - 1}),$ where $X^{i} = (X_{1}, \dots, X_{i})$ and $Y^{0}$ is the empty tuple. For a third finite sequence $Z^{n}$ , define the causally conditioned version $I (X^{n} \to Y^{n} ∥ Z^{n}) : = \sum_{i = 1}^{n} I (X^{i}; Y_{i} ∣ Y^{i - 1}, Z^{i}) .$ All conditional mutual information terms use the finite nats convention of Finite conditional mutual information. The conditioning includes $Z_{i}$ because it is the current side-information symbol available at the receiving end of the directed-information term; $Y^{i - 1}$ and $Z^{i - 1}$ are the past.

Depends on

Finite conditional mutual information

Used by

Mutual information splits into directed information and feedback Lemma

Dependency tree · two levels

6 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

Abbas El Gamal and Young-Han Kim, Lecture Notes on Network Information Theory, arXiv:1001.3404v4 (standard reference, not scraped)