ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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.

The same Bernoulli law with two different targets

Example

Let $X = {0, 1}$ with its power-set sigma-algebra and $Θ = [0, 1]^{2}$ . For $(u, v) \in Θ$ , let $P_{(u, v)}$ be the Bernoulli law with $P_{(u, v)} ({1}) = u$ and $P_{(u, v)} ({0}) = 1 - u$ , and take the target $ψ (u, v) = v$ with its usual metric. The two parameters $θ_{0} = (1 / 2, 0)$ and $θ_{1} = (1 / 2, 1)$ have the same observation law and different target values. Thus the target is not identifiable and no estimator sequence is consistent in probability at both parameters.

Facts & Assumptions

Given: The finite observation space, model family, target and two parameters in the example.

[F1]

Sigma-algebras: The power set of a set is a sigma-algebra.

[F2]

Measurable spaces and measurable sets: A measurable space is a set equipped with a sigma-algebra.

[F3]

Measures on sigma-algebras: A measure is countably additive on pairwise disjoint measurable sets and assigns measure zero to the empty set.

[F4]

Probability measures and probability spaces: A probability measure is a measure with total mass one.

[F5]

Statistical experiments: An experiment is a measurable observation space and a parameter-indexed family of probability measures on it.

[F6]

Parametric statistical models: A parametric model is the map from parameter labels to their sampling laws; it need not be injective.

[F7]

Estimands on a statistical model: An estimand is a specified target-valued function on the parameter set.

[F8]

Observational equivalence: Two parameters are observationally equivalent exactly when their probability measures on the observation space are equal.

[F9]

Identifiability of an estimand: An estimand is identifiable when observationally equivalent parameters have equal target values.

[F10]

The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded: The usual distance on the real target satisfies the triangle inequality.

[F11]

Estimator sequences in IID experiments: An estimator sequence is a measurable map on each finite IID product sample space.

[F12]

Pointwise consistency in probability: Pointwise consistency at a parameter means the estimation-error probability under its IID sample laws tends to zero for every positive tolerance.

[F13]

Nonidentifiability precludes simultaneous consistency: If two observationally equivalent parameters have distinct targets in a metric space, no estimator sequence can be consistent at both.

[F14]

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique: The finite product measure assigns a measurable rectangle the product of its factor probabilities.

Verification

1.1F1F2F3F4F5F6F7given

The power set on $X$ is a sigma-algebra by [F1], so $(X, P (X))$ is a measurable space by [F2]. For every $u \in [0, 1]$ , the two singleton masses $u$ and $1 - u$ are nonnegative and sum to one. Any disjoint sequence of subsets of $X$ has at most two nonempty members, so the assignment is countably additive and gives zero to the empty set by [F3]; hence each $P_{(u, v)}$ is a probability measure by [F4]. This defines the experiment and model in [F5]–[F6], and $ψ$ is the target in [F7].

1.2F7F8F9F10givenalgebra

At $θ_{0} = (1 / 2, 0)$ and $θ_{1} = (1 / 2, 1)$ , both laws assign probability $1 / 2$ to each singleton and therefore are equal as measures on the power set. The target values are $0$ and $1$ , whose distance is $1 > 0$ . Thus [F8] makes the parameters observationally equivalent, while [F7] and [F9] show the target is not identifiable.

2.1F11F12F13F14givenstep 1.2∎

By [F11]–[F12], estimators are measurable maps on the IID sample spaces and consistency is pointwise under those sample laws. Apply [F13] to the pair from step 1.2: no estimator sequence is consistent in probability at both $θ_{0}$ and $θ_{1}$ . For the finite check at sample size $n = 2$ , [F14] gives mass $(1 / 2)^{2} = 1 / 4$ to each ordered pair under both product laws, confirming that the observation distributions remain identical.

Depends on

Used by

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Sources

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 15, PDF pp. 6–8 (standard reference, not scraped)
John Duchi, Theory of Statistics, Stats 300B Lecture 3, Winter 2018, PDF pp. 3–4 (standard reference, not scraped)