ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Perfectly correlated inputs share a Shapley effect

Example

Assume AC. Let $Z$ be symmetric on ${- 1, 1}$ , set $X_{1} = X_{2} = Z$ , and let $Y = f (X_{1}, X_{2}) = Z$ . Then $V = Var (Y) = 1$ . The independent-input Hoeffding decomposition is not applicable because $X_{1}$ and $X_{2}$ are perfectly dependent. Under the Shapley convention of Shapley effects for dependent inputs, both inputs receive effect $1 / 2$ .

Facts & Assumptions

Given: AC, symmetric $Z \in {- 1, 1}$ , $X_{1} = X_{2} = Z$ , and $Y = Z$ .

[A1]

The Axiom of Choice: AC is assumed for the conditional-expectation projection supplier.

[F1]

Shapley effects for dependent inputs: $v (S) = Var (E [Y ∣ X_{S}])$ and each effect averages predecessor-set increments over permutations.

[F2]

Dependent-input Shapley effects are nonnegative and sum to one: the finite permutation effects are nonnegative and sum to one.

[F3]

Exact finite-permutation Shapley effects: the procedure enumerates every permutation, adds each predecessor-set marginal to the corresponding accumulator, and returns the accumulators divided by $d! V$ .

[F4]

Moments, variance, and covariance on a probability space: $Var (Z) = E [(Z - E Z)^{2}]$ .

[F5]

Conditional expectation is the l2 orthogonal projection: under AC, conditional expectation fixes variables measurable with respect to the conditioning sigma-algebra.

Verification

Given: The model and conventions in the Example.

1.1A1F1F4F5givenalgebra

Symmetry gives $E Z = 0$ and $E Z^{2} = 1$ , so $V = 1$ . Since $Y$ is measurable with respect to either $X_{1}$ or $X_{2}$ , [F5] fixes its conditional expectation for either singleton, giving $v (\emptyset) = 0$ , $v ({1}) = v ({2}) = 1$ , and $v ({1, 2}) = 1$ .

2.1F1F3step 1.1algebra

For permutation $(1, 2)$ , the increments are $1 - 0 = 1$ for input 1 and $1 - 1 = 0$ for input 2. For permutation $(2, 1)$ , they are $1$ for input 2 and $0$ for input 1. There are exactly these two permutations, so [F3] returns $ϕ_{1} = ϕ_{2} = (1 + 0) / (2 V) = 1 / 2$ .

3.1F2step 2.1∎

Both effects are nonnegative and $ϕ_{1} + ϕ_{2} = 1$ , in agreement with [F2]. The allocation splits variance that either input fully reveals; it does not assert an independent-input Sobol decomposition.

Depends on

Used by

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Dependency tree · two levels

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Sources

Eunhye Song, Barry L. Nelson, and Jeremy Staum, Shapley Effects for Global Sensitivity Analysis: Theory and Computation, §§2–3 (standard reference, not scraped)