DefinitionDefinition: AI-adaptedProof: 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.

Product-law Hoeffding ANOVA components

Definition

Let $f \in L^{2} (μ)$ on the finite product probability space of Product-coordinate averaging operators. For $u \subseteq [d]$ , its Hoeffding ANOVA component indexed by $u$ is the $L^{2}$ class $f_{u} : = \sum_{v \subseteq u} (- 1)^{∣ u ∣ - ∣ v ∣} P_{v} f .$ In particular $f_{\emptyset} = P_{\emptyset} f = E f$ . This convention includes all interaction orders up to $d$ ; $f_{u}$ depends only on coordinates in $u$ once the coordinate averages are identified with their lifts to the full product.

Depends on

Used by

Dependency tree · two levels

7 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

Thierry Klein, Introduction to Sensitivity Analysis: The Statistical Study of Sobol Indices, complete lecture notes, §2.2.2 (standard reference, not scraped)