LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

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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.

Permutation-equivariant scores preserve exchangeability

Statement

Let $Z = (Z_{1}, \dots, Z_{n})$ be an exchangeable random element of $Z^{n}$ , and let $A : Z^{n} \to R^{n}$ be measurable and permutation-equivariant in the sense of Conformal scores and rank p-values. Then $A (Z)$ is an exchangeable real random vector.

Facts & Assumptions

Given: The exchangeable data tuple and measurable permutation-equivariant score assignment in the statement.

[F1]

Exchangeability means the law is invariant under every coordinate permutation. Exchangeable random elements

[F2]

The score assignment satisfies $A (τ_{π} z) = τ_{π} A (z)$ for every permutation. Conformal scores and rank p-values

[F3]

Coordinate permutations on finite product measurable spaces are measurable. Coordinate maps and permutations are measurable on finite products

Proof

1.1givenF2F3

The measurable map $A$ sends $Z$ to a random vector by the composition rule for measurable maps; [F3] makes every coordinate permutation of that vector measurable, so its law is defined.

2.1F1F2step 1.1∎

For each permutation $π$ , [F2] gives $τ_{π} A (Z) = A (τ_{π} Z)$ . By exchangeability [F1], $τ_{π} Z$ and $Z$ have the same law; applying the measurable map $A$ shows that $A (τ_{π} Z)$ and $A (Z)$ have the same law. Hence $A (Z)$ is exchangeable.

Depends on

Used by

Full conformal marginal coverage Theorem
Split conformal marginal coverage Theorem

Dependency tree · two levels

12 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

Jing Lei et al., Distribution-Free Predictive Inference for Regression, JASA 113(523) (2018), §2.1 and Remark 2.4 (standard reference, not scraped)
Ryan Tibshirani, Conformal Prediction, Advanced Topics in Statistical Learning, Spring 2024, §3 (standard reference, not scraped)