TheoremStatement: Literature-sourcedProof: 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.

Mean-zero score and the information identity

Statement

Assume the Axiom of Choice for the selected density representatives. Let $Θ$ be open in $R^{d}$ , let $μ$ be a fixed sigma-finite dominating measure, and let $p_{η}$ be the selected densities of a statistical model. Fix $θ \in Θ$ . Suppose that on a neighborhood $U$ of $θ$ there are a fixed measurable support $S$ and a $μ$ -null set $N$ such that, for every $x \in S ∖ N$ , $p_{η} (x) > 0$ and $η \mapsto p_{η} (x)$ has continuous first and second partial derivatives on $U$ ; outside $S \cup N$ , $p_{η} (x) = 0$ for all $η \in U$ . Suppose there are nonnegative $g_{1}, g_{2} \in L^{1} (μ)$ that uniformly dominate every first partial derivative and every second partial derivative, respectively, on $U$ . Also assume $E_{θ} ∥ S_{θ} (X) ∥^{2} < \infty$ .

Then $E_{θ} S_{θ} (X) = 0$ , and, entrywise, $I (θ) = E_{θ} [S_{θ} (X) S_{θ} (X)^{T}] = - E_{θ} [\nabla_{θ}^{2} ℓ_{θ} (X)] .$ For a scalar parameter this reads $E_{θ} S_{θ} = 0$ and $I (θ) = E_{θ} S_{θ}^{2} = - E_{θ} [ℓ_{θ}^{''} (X)]$ .

This common-support, integrable-envelope setup is sufficient, not necessary: the theorem does not claim that every model with valid score identities must have common support or these envelopes.

Facts & Assumptions

Given: AC; $Θ \subseteq R^{d}$ open; a fixed sigma-finite dominating measure $μ$ ; a selected density family $p_{η}$ ; a fixed support $S$ and common null set $N$ on a neighborhood $U$ ; pointwise continuous first and second partial derivatives on $S ∖ N$ ; integrable uniform envelopes $g_{1}, g_{2}$ for those derivatives; and a finite second moment of the score.

[A1]

The Axiom of Choice: AC says every family of nonempty sets has a choice function; here it supplies the simultaneous family of density representatives.

[F1]

Dominated statistical experiments: every sampling law $P_{η}$ is dominated by the fixed sigma-finite positive measure $μ$ .

[F2]

Likelihood densities: $p_{η}$ is a selected measurable Radon–Nikodym representative of $d P_{η} / d μ$ .

[F3]

A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density: $P_{η} (A) = \int_{A} p_{η} d μ$ for every measurable $A$ .

[F4]

Probability measures and probability spaces: $P_{η} (X) = 1$ .

[F5]

The measure with density $f$ relative to $μ$ and Integrating against a density agrees with integrating the product: if $P_{η} = p_{η} d μ$ , then $E_{η} f (X) = \int f p_{η} d μ$ for nonnegative $f$ . For real integrable $f$ , apply this identity to $f^{+}$ and $f^{-}$ ; applying it first to $∣ f ∣$ shows $∣ f ∣ p_{η}$ is integrable, so the signed identity follows from the definition of the signed integral.

[F6]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : for a differentiable function on an interval, the difference quotient between two endpoints equals its derivative at an intermediate point.

[F7]

Dominated convergence: pointwise almost-everywhere convergence under one integrable majorant implies convergence of the integrals.

[F8]

Score and Fisher information: on $p_{θ} > 0$ , the score is $\nabla_{θ} \log p_{θ}$ and the information matrix has entries $E_{θ} [S_{θ, i} S_{θ, j}]$ .

[F9]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: on the positive support, $\partial_{j} ℓ = (\partial_{j} p) / p$ and $\partial_{i j} ℓ = (\partial_{i j} p) / p - (\partial_{i} p) (\partial_{j} p) / p^{2}$ .

[F10]

Expectation of a nonnegative or integrable random variable and Integrable real and complex functions, and their integrals: expectations are integrals and signed expectations require integrability.

[F11]

The Lebesgue integral is linear on $L^{1} (μ)$ : for integrable functions, the integral commutes with differences and scalar multiplication.

Proof

technique · direct

1.1A1F1F2F3F4F5F6F7F8F11givenchoose

By [F2]–[F4], every selected density has $\int p_{η} d μ = P_{η} (X) = 1$ . Fix a coordinate $j$ and a sequence $h_{n} \to 0$ with $θ + h_{n} e_{j} \in U$ . For almost every $x \in S$ , the mean value theorem [F6] bounds $∣ (p_{θ + h_{n} e_{j}} (x) - p_{θ} (x)) / h_{n} ∣$ by $g_{1} (x)$ , and the quotient converges to $\partial_{j} p_{θ} (x)$ . Dominated convergence [F7] therefore gives convergence of the quotient integrals. Linearity [F11] shows each quotient integral is $(1 - 1) / h_{n} = 0$ , so $\int \partial_{j} p_{θ} d μ = 0$ . Since $E_{θ} S_{θ, j}^{2} < \infty$ and $P_{θ}$ is a probability, $S_{θ, j}$ is integrable (use $∣ z ∣ \leq (1 + z^{2}) / 2$ ). On $S$ , the signed density identity [F5] and [F8] identify $E_{θ} S_{θ, j}$ with this integral; outside $S$ the law is zero. Thus every score coordinate has mean zero.

2.1F6F7F11step 1.1given

For any coordinates $i, j$ , apply the same sequence argument to $\partial_{i} p_{θ + h_{n} e_{j}}$ . The mean value theorem bounds its difference quotient by $g_{2}$ , so [F7] interchanges this derivative and the integral. Step 1.1 holds at every parameter in $U$ , hence $\int \partial_{i} p_{η} d μ = 0$ throughout $U$ ; linearity [F11] turns each difference-quotient integral into the difference of these zero integrals. Dominated convergence then gives $\int \partial_{i j} p_{θ} d μ = 0$ .

3.1F5F8F9F10F11step 2.1givenalgebra

On $S ∖ N$ , differentiate $ℓ_{θ} = \log p_{θ}$ using [F9]. Multiplying the Hessian formula by $p_{θ}$ , integrating with the signed density identity [F5] and linearity [F11], and using step 2.1 yields $E_{θ} [\partial_{i j} ℓ_{θ}] = \int \partial_{i j} p_{θ} d μ - E_{θ} [S_{θ, i} S_{θ, j}] = - E_{θ} [S_{θ, i} S_{θ, j}]$ . The terms are integrable: $\int ∣ \partial_{i j} p_{θ} ∣ d μ \leq \int g_{2} d μ$ , and $2 ∣ S_{θ, i} S_{θ, j} ∣ \leq S_{θ, i}^{2} + S_{θ, j}^{2}$ .

4.1step 1.1step 3.1∎

The coordinate identities in step 3.1 give the asserted matrix identity; when $d = 1$ they give the scalar identity. Step 1.1 supplies the score mean-zero claim in both cases.

Depends on

Used by

Uniform moving support invalidates the naive score bound Counterexample
Bernoulli score, information, and attainment of the scalar bound Example
Normal location score, information, and attainment Example
Multivariate Cramér–Rao bound Theorem
Scalar Cramér–Rao lower bound Theorem

Dependency tree · two levels

65 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

George Deligiannidis, SB2.1 Foundations of Statistical Inference, §3.1–3.2, Theorems 3.4 and 3.7, PDF pp. 21–24 (standard reference, not scraped)
Peter Kempthorne, MIT 18.655 Lecture 13: Unbiased Estimation and Risk Inequalities, PDF pp. 6–8, 16–17 (standard reference, not scraped)