LemmaStatement: AI-adaptedProof: 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.

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Nested-model deviance is nonnegative

Statement

For the deviance in Likelihood deviance for nested models on common support, $D (y) \geq 0$ .

Facts & Assumptions

Given: Nonempty nested parameter sets $Θ_{f i t} \subseteq Θ_{s a t}$ , a common log likelihood $ℓ (β; y)$ , and finite real suprema.

[F1]

Likelihood deviance for nested models on common support: $D (y) = 2 {\sup_{Θ_{s a t}} ℓ (β; y) - \sup_{Θ_{f i t}} ℓ (β; y)}$ and $Θ_{f i t} \subseteq Θ_{s a t}$ .

Proof

1.1F1given

Every value $ℓ (β; y)$ with $β \in Θ_{f i t}$ is also among the values over $Θ_{s a t}$ by [F1]. Therefore $\sup_{Θ_{f i t}} ℓ (β; y) \leq \sup_{Θ_{s a t}} ℓ (β; y)$ .

2.1F1step 1.1algebra∎

The suprema are finite real numbers by the definition's hypotheses, so their ordered difference is nonnegative; multiplying by $2 > 0$ gives $D (y) \geq 0$ .

Depends on

Likelihood deviance for nested models on common support

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

MIT 18.650 Statistics for Applications, Lecture 10: GLM likelihood background; the inequality is proved locally (standard reference, not scraped)