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.

Exchangeable random elements

Definition

Let $Z_{1}, \dots, Z_{n}$ be random elements, $n \geq 1$ , of one measurable space $(Z, Z)$ . Their tuple $Z = (Z_{1}, \dots, Z_{n})$ is a random element of $(Z^{n}, Z^{\otimes n})$ by Coordinate maps and permutations are measurable on finite products, and its law is the pushforward probability measure from Law or distribution of a random element. For a permutation $π$ of ${1, \dots, n}$ , let $τ_{π} (z)_{i} = z_{π (i)}$ . The tuple is exchangeable if $P_{Z} \circ τ_{π}^{- 1} = P_{Z} for every permutation π .$

This is invariance of the joint law, not an assertion of independence or identical marginal construction by independent sampling. The coordinate maps $τ_{π}$ are measurable by the cited finite-product lemma.

Depends on

Used by

Conformal scores and rank p-values Definition
Full conformal ranks a candidate in an augmented sample Example
Permutation-equivariant scores preserve exchangeability Lemma
Conformal rank superuniformity under exchangeability Theorem
Full conformal marginal coverage Theorem
Split conformal marginal coverage Theorem

Dependency tree · two levels

11 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

Glenn Shafer and Vladimir Vovk, A Tutorial on Conformal Prediction, JMLR 9 (2008), §3.1 (standard reference, not scraped)
Ryan Tibshirani, Conformal Prediction, Advanced Topics in Statistical Learning, Spring 2024, §1 (standard reference, not scraped)