TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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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General-link score and Fisher information

Statement

Consider Generalized linear model with explicit conditioning at an admissible finite $β$ with common scale $a (ϕ) > 0$ . Assume each $η_{i} \mapsto θ_{i}$ is differentiable, each $η_{i} \mapsto μ_{i}$ is differentiable with ${\dot{μ}}_{i} : = d μ_{i} / d η_{i} \neq 0$ , $V_{i} : = b^{''} (θ_{i}) > 0$ , the component scores have finite second moments, and the common-support, differentiability, and local integrable derivative-envelope conditions from Explicit exponential-family response contracts hold. Put

$w_{i} = \frac{{\dot{μ}}_{i}^{2}}{a (ϕ) V_{i}}, z_{i} = η_{i} + \frac{y_{i} - μ_{i}}{{\dot{μ}}_{i}} .$

For every observed $y$ with $f_{i} (y_{i}; θ_{i}, ϕ) > 0$ for each $i$ , the score and expected score outer-product information are

$U (β; y, ϕ) = X^{T} {(\frac{{\dot{μ}}_{i} (y_{i} - μ_{i})}{a (ϕ) V_{i}})}_{i = 1}^{n}, I (β, ϕ) = X^{T} diag (w_{i}) X .$

In particular, if $X$ has full column rank, then $I$ is positive definite.

Facts & Assumptions

Given: The finite independent-response GLM, common scale, differentiable link and component natural-parameter maps, an observed response vector with positive component densities, fixed common support, local derivative envelopes, and finite second moments in the Statement.

[F1]

Generalized linear model with explicit conditioning: $μ_{i} = g^{- 1} (η_{i})$ , $η_{i} = x_{i}^{T} β$ , the score is the gradient of the product log likelihood, and $I = E [U U^{T}]$ when finite.

[F2]

Explicit exponential-family response contracts: the component responses are independent, have the displayed common-support response densities, and use the stated product sampling law.

[F3]

Mean and variance from the cumulant function: $E [Y_{i}] = μ_{i}$ and $Var (Y_{i}) = a (ϕ) V_{i}$ under the common-scale restriction.

[F4]

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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : differentiating a composition multiplies the outer and inner derivatives under its stated hypotheses.

[F5]

Expectations factor over finite products of independent random variables: for independent real random variables, expectations of products of integrable measurable coordinate functions factor.

Proof

1.1F1F3F4algebra

Differentiate $μ_{i} = b^{'} (θ_{i})$ with respect to $η_{i}$ . The chain rule [F4] gives ${\dot{μ}}_{i} = V_{i} d θ_{i} / d η_{i}$ , so $d θ_{i} / d η_{i} = {\dot{μ}}_{i} / V_{i}$ . Differentiating the component log density $(y_{i} θ_{i} - b (θ_{i})) / a (ϕ) + c_{i} (y_{i}, ϕ)$ therefore gives the scalar score $s_{i} = {\dot{μ}}_{i} (y_{i} - μ_{i}) / (a (ϕ) V_{i})$ . Since $η_{i} = x_{i}^{T} β$ , the full score is $U = \sum_{i} x_{i} s_{i} = X^{T} (s_{i})_{i = 1}^{n}$ .

2.1F1F2F3F5step 1.1algebra∎

By [F3], $E [s_{i}] = 0$ and $E [s_{i}^{2}] = {\dot{μ}}_{i}^{2} / (a (ϕ) V_{i}) = w_{i}$ . These second moments are finite by the hypotheses, and $∣ s_{i} ∣ \leq (1 + s_{i}^{2}) / 2$ makes each $s_{i}$ integrable. For $i \neq j$ , independence and [F5] give $E [s_{i} s_{j}] = E [s_{i}] E [s_{j}] = 0$ . Expanding $E [U U^{T}]$ now leaves only $\sum_{i} x_{i} x_{i}^{T} w_{i} = X^{T} diag (w_{i}) X$ . If $v \neq 0$ and $X$ has full column rank, then $X v \neq 0$ and $\sum_{i} w_{i} (x_{i}^{T} v)^{2} > 0$ because each $w_{i} > 0$ , proving positive definiteness.

Depends on

Used by

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Sources

Philippe Rigollet, MIT 18.650 Statistics for Applications, Lecture 10: Generalized Linear Models, PDF pp. 43–48 (standard reference, not scraped)
Ryan Tibshirani, Generalized Linear Models, CMU Advanced Methods for Data Analysis, PDF pp. 4–5 (standard reference, not scraped)