DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedprecheck pendingjudge 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.

Minimal sufficient sigma-fields

Definition

For a family $(P_{θ})_{θ \in Θ}$ on $(X, A)$ , a sub- $σ$ -field $G \subseteq A$ is sufficient if for every bounded measurable $u : X \to R$ there is one $G$ -measurable bounded function $v_{u}$ such that

$\int_{A} u d P_{θ} = \int_{A} v_{u} d P_{θ} (A \in G, θ \in Θ) .$

For each $θ$ , this says that $v_{u}$ is a version of $E_{θ} [u ∣ G]$ , with the same pointwise version used for every parameter. Define the family-null ideal $N = {N \in A : P_{θ} (N) = 0 for every θ}$ . We say $G \subseteq H (m o d N)$ if for every $A \in G$ there is $B \in H$ with $A △ B \in N$ . A sufficient $σ$ -field $G_{*}$ is minimal sufficient if $G_{*} \subseteq H (m o d N)$ for every sufficient $H$ .

Depends on

Used by

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

Ryan Martin, Lecture Notes on Advanced Statistical Theory (standard reference, not scraped)
Joshua M. Tebbs, STAT 512 Mathematical Statistics Lecture Notes (standard reference, not scraped)
Peter Kempthorne, MIT 18.655 Lecture Note 6: Sufficiency (standard reference, not scraped)
Noah Fithian, Stat 210a Reader: Sufficiency (standard reference, not scraped)
Noah Fithian, Stat 210a Reader: Completeness, Ancillarity, and Basu’s Theorem (standard reference, not scraped)
Charles J. Geyer, Stat 5421 Lecture Notes: Exponential Families (standard reference, not scraped)
Jiahao Tian, Chapter 2: Sufficiency and Exponential Families (standard reference, not scraped)