ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

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Two endpoint laws dominate the Bernoulli scale family

Example

Let $X = {0, 1}$ and $P_{θ} = Bernoulli (θ)$ for $0 \leq θ \leq 1$ . The positive mixture $Q = \frac{1}{2} P_{0} + \frac{1}{2} P_{1}$ dominates every $P_{θ}$ .

Facts & Assumptions

Given: the two-point sample space and the Bernoulli family including both endpoints.

[F1]

Probability measures and probability spaces: each model member and the displayed mixture is a probability measure on the two-point space.

[F2]

The measure with density $f$ relative to $μ$ : a nonnegative function $q$ defines the measure $q d Q$ by $A \mapsto \int_{A} q d Q$ .

Proof

1.1F1givenalgebra

The endpoint laws are $P_{0} = δ_{0}$ and $P_{1} = δ_{1}$ , so $Q = \frac{1}{2} δ_{0} + \frac{1}{2} δ_{1}$ assigns mass $1 / 2$ to each point. This is a probability measure.

2.1F1F2step 1.1givenalgebra∎

For every $A \subseteq {0, 1}$ , if $Q (A) = 0$ then $A = \emptyset$ , so $P_{θ} (A) = 0$ ; hence $P_{θ} ≪ Q$ . Define $q_{θ} (0) = 2 (1 - θ)$ and $q_{θ} (1) = 2 θ$ . For each subset $A$ , $\int_{A} q_{θ} d Q = 1_{{0 \in A}} (1 - θ) + 1_{{1 \in A}} θ = P_{θ} (A)$ . Thus $q_{θ} = d P_{θ} / d Q$ , including its zero values at endpoints, and the displayed finite mixture directly dominates every member.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)
Jiahao Tian, Chapter 2: Sufficiency and Exponential Families (standard reference, not scraped)