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.

Marginal score function

Definition

Let $d \geq 1$ be an integer, let $I \subseteq R$ be a time interval, and let $P_{t}$ be a probability law on $R^{d}$ for each $t \in I$ . Suppose $P_{t}$ has a density $p_{t}$ with respect to Lebesgue measure, meaning $P_{t} (A) = \int_{A} p_{t} (x) d x$ for every Borel set $A \subseteq R^{d}$ . On the set of points $x$ where $0 < p_{t} (x) < \infty$ and the map $x \mapsto p_{t} (x)$ is differentiable, the marginal score of $P_{t}$ is

$s_{t} (x) : = \nabla_{x} \log p_{t} (x) .$

The score is a spatial vector field; this definition asserts no value where the density is zero, infinite, or not differentiable.

Definition references

Song et al., Section 3.3

Raginsky, Section 7.2

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Yang Song et al., Score-Based Generative Modeling through Stochastic Differential Equations, Section 3.3 (standard reference, not scraped)
Maxim Raginsky, Stochastic Differential Equations: A Systems-Theoretic Approach, Section 7.2 (standard reference, not scraped)