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.

Density-defined Jensen–Shannon divergence

Definition

In the common-density setting, put $m = (p + q) / 2$ . Additivity and nonnegative homogeneity of the integral give $\int m d μ = 1$ . Define $r = p / m$ and $s = q / m$ where $m > 0$ , and define $r = s = 0$ where $m = 0$ ; then $0 \leq r, s \leq 2$ and $r + s = 2$ wherever $m > 0$ . Let $ϕ (0) = 0$ and $ϕ (t) = t \ln t$ for $t > 0$ . The density-defined Jensen–Shannon divergence is

{JS}_{μ} (P, Q) : = \frac{1}{2} \int m ϕ (r) d μ + \frac{1}{2} \int m ϕ (s) d μ .

The integrals are finite, and this formula is the average of the two density-relative-entropy terms from $p$ and $q$ to the mixture density $m$ .

Definition references

Common-density setting and probability laws

Lebesgue integrability of signed functions

Natural logarithm

Coordinatewise measurability into Euclidean space

Composition with a Borel map

Measurability of arithmetic and products

Additivity of the nonnegative integral

Nonnegative homogeneity of the integral

Goodfellow et al., §4.1, equations (5)–(6), complete 9-page paper

Li, §1.2, equation (11), complete 6-page lecture note

Depends on

Used by

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Sources

Ian J. Goodfellow et al., Generative Adversarial Nets, arXiv:1406.2661v1 (standard reference, not scraped)
Yingzhen Li, Lecture notes on generative adversarial networks (GANs), Deep Learning (Winter 2022, Imperial College London) (standard reference, not scraped)