AlgorithmAlgorithm: 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.

Exact finite-permutation Shapley effects

Definition

Input: A finite player set $[d]$ , an exact oracle for $v (S)$ at every $S \subseteq [d]$ , and a known $V > 0$ satisfying $v (\emptyset) = 0$ and $v ([d]) = V$ . In the sensitivity application, $v$ is the conditional-projection variance game of Shapley effects for dependent inputs.

Initialize $A_{i} = 0$ for every $i \in [d]$ .
For each of the $d!$ permutations $π$ , traverse its entries in order. Start with $S = \emptyset$ . At entry $i$ , add $v (S \cup {i}) - v (S)$ to $A_{i}$ and replace $S$ by $S \cup {i}$ .
Return $ϕ_{i} = A_{i} / (d! V)$ for all $i$ .

The domain excludes $V = 0$ and unavailable coalition values; the procedure does not approximate conditional expectations or their numerical errors.

Depends on

Shapley effects for dependent inputs

Used by

Dependency tree · two levels

4 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

Eunhye Song, Barry L. Nelson, and Jeremy Staum, Shapley Effects for Global Sensitivity Analysis: Theory and Computation, §3.1 (standard reference, not scraped)