DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Finite Bradley–Terry preference model

Definition

Let $X$ be a finite nonempty prompt set. For each $x \in X$ , let $Y_{x}$ be a finite response set with at least two elements, and let $r_{x} : Y_{x} \to R$ be a real-valued reward. The Bradley–Terry model assigns to each ordered pair of distinct responses $a, b \in Y_{x}$ the two-outcome comparison $a ≻ b$ or $b ≻ a$ , with $\Pr (a ≻ b ∣ x) = σ (r_{x} (a) - r_{x} (b)), σ (t) = \frac{1}{1 + e^{- t}} .$ Here $a ≻ b$ means that response $a$ is preferred to response $b$ at prompt $x$ ; the probability of the reverse outcome is $1 - \Pr (a ≻ b ∣ x) = σ (r_{x} (b) - r_{x} (a))$ . This is a statistical model for pairwise comparisons, not a universal law of human preference.

Definition references

Binary logistic and multiclass softmax models: defines the same sigmoid $σ (t) = (1 + e^{- t})^{- 1}$ for finite real $t$ .

Rafailov et al., §3, Bradley–Terry preference model

Purdue PrefOpt course notes, §2, Bradley–Terry model

Depends on

Binary logistic and multiclass softmax conditional models

Used by

Dependency tree · two levels

2 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

Rafael Rafailov et al., Direct Preference Optimization: Your Language Model is Secretly a Reward Model, arXiv:2305.18290v3 (2024) (standard reference, not scraped)
Purdue course lecture notes, Preference Optimization (PrefOpt) (standard reference, not scraped)