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.

The ancestral moral graph of a DAG

Definition

For $S \subseteq V$ , its ancestral set is ${An}_{G} (S) : = {v \in V : there is a directed path from v to some s \in S},$ where paths of length zero are allowed, so $S \subseteq {An}_{G} (S)$ . The induced directed subgraph on this set keeps exactly the arrows of $G$ whose two endpoints lie in ${An}_{G} (S)$ . Its moral graph is the undirected simple graph obtained by replacing every arrow by an undirected edge and joining every distinct pair of parents of a common child by an undirected edge; loops and duplicate edges are discarded. This undirected graph is the ancestral moral graph on $S$ and is denoted ${mor}_{G} (S)$ .

Depends on

Paths, active paths, and d-separation in a finite DAG

Used by

Dependency tree · one level

1 result within one dependency step 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

Judea Pearl, Causal Diagrams for Empirical Research, §2-1, p. 672 (standard reference, not scraped)
Miguel A. Hernán and James M. Robins, Causal Inference: What If, Fine Point 6.1, pp. 78–79 (standard reference, not scraped)