ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge 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.

Identifiable Bernoulli model and parameter target

Example

Let $X = {0, 1}$ with $A = 2^{X}$ , let $Θ = [0, 1]$ , and, for $p \in Θ$ , define

P_{p} (A) : = p 1_{{1 \in A}} + (1 - p) 1_{{0 \in A}} (A \in A),

where each indicator is $1$ when its stated membership holds and $0$ otherwise. This is the Bernoulli model with success probability $p$ . Its parameter is identifiable, and the target $ψ (p) = p$ is identifiable.

Facts & Assumptions

Given: The parameter space is $[0, 1]$ and the sample space is ${0, 1}$ .

[F1]

Sigma-algebras: a sigma-algebra contains the whole space and is closed under complements and countable unions; the full power set satisfies these conditions.

[F2]

Measurable spaces and measurable sets: a measurable space is a set together with a sigma-algebra on it.

[F3]

Probability measures and probability spaces: a probability measure is a measure of total mass $1$ on a measurable space.

[F4]

Statistical experiments: an experiment assigns a probability measure $P_{p}$ on the sample measurable space to each parameter $p$ .

[F5]

Parametric statistical models: the model map sends $p$ to its sampling law $P_{p}$ .

[F6]

Estimands on a statistical model: $ψ (p) = p$ is a function from the parameter set to the target set $[0, 1]$ .

[F7]

Identifiability of an estimand: identifiability means equal sampling laws imply equal target values.

[F8]

Identifiability is constancy on observational classes: an identifiable target factors uniquely through the set of sampling laws.

Verification

Given: The finite Bernoulli family and target specified above.

1.1givenF1F2F3F4algebra

The full power set contains $X$ and is closed under finite unions and complements, so it is an algebra of subsets. Every countable union of its members is still a subset of $X$ , so it is a sigma-algebra by [F1] and gives a measurable space by [F2]. For any $p \in [0, 1]$ , both singleton masses $p$ and $1 - p$ are nonnegative, and $P_{p} (X) = p + (1 - p) = 1$ . If $(A_{n})$ is pairwise disjoint, each of $0$ and $1$ belongs to at most one $A_{n}$ , so $\sum_{n} P_{p} (A_{n}) = p 1_{{1 \in \cup_{n} A_{n}}} + (1 - p) 1_{{0 \in \cup_{n} A_{n}}} = P_{p} (\cup_{n} A_{n})$ . Hence every $P_{p}$ is a probability measure by [F3].

2.1F5F6F7F8algebra∎

If $P_{p} = P_{p^{'}}$ , evaluation on ${1}$ gives $p = P_{p} ({1}) = P_{p^{'}} ({1}) = p^{'}$ . Thus the model map is injective and equal laws force $ψ (p) = ψ (p^{'})$ ; the target is identifiable by [F7]. By [F8] it factors uniquely through the set of Bernoulli laws.

Depends on

Used by

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Dependency tree · two levels

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