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.

Source and target joint laws, risks, and covariate shift

Definition

Let $(X, Σ_{X})$ , $(Y, Σ_{Y})$ , and $(A, Σ_{A})$ be nonempty measurable spaces. A source/target pair of joint laws consists of probability measures $P_{S}$ and $P_{T}$ on $(X \times Y, Σ_{X} \otimes Σ_{Y})$ . For $D \in {S, T}$ , its input marginal is the probability measure $μ_{D}$ on $(X, Σ_{X})$ defined by $μ_{D} (A) = P_{D} (A \times Y)$ . If $h : X \to A$ is measurable and $ℓ : A \times Y \to [0, + \infty]$ is product-measurable, then $(x, y) \mapsto (h (x), y)$ is measurable: the preimage of a measurable rectangle $U \times V$ is $h^{- 1} (U) \times V$ , and such rectangles generate the product sigma-algebra. Taking preimages once more shows that $(x, y) \mapsto ℓ (h (x), y)$ is measurable. The domain risk is the extended nonnegative integral $R_{D} (h) : = \int_{X \times Y} ℓ (h (x), y) P_{D} (d (x, y)) .$

A conditional label kernel for $P_{D}$ is a specified probability kernel $K_{D}$ from $(X, Σ_{X})$ to $(Y, Σ_{Y})$ such that, for every nonnegative product-measurable $g : X \times Y \to [0, + \infty]$ , $\int g (x, y) P_{D} (d (x, y)) = \int_{X} (\int_{Y} g (x, y) K_{D} (x, d y)) μ_{D} (d x) .$ The pair is under covariate shift when conditional label kernels $K_{S}, K_{T}$ are supplied, there is a measurable set $N$ with $μ_{T} (N) = 0$ such that $K_{S} (x, \cdot) = K_{T} (x, \cdot)$ as probability measures for every $x \notin N$ , and $μ_{T} ≪ μ_{S}$ . No existence of conditional kernels is asserted by this definition.

Definition references

Supervised prediction problems: gives the measurable spaces, joint probability law, predictor class, and measurable loss convention.

Population and empirical risk: defines domain risk as an extended nonnegative integral.

A measurable function between measurable spaces: a map is measurable when preimages of measurable sets are measurable.

The product sigma-algebra and its finite iterates: the product sigma-algebra is generated by measurable rectangles.

Measure kernel and probability kernel: a probability kernel assigns a probability measure to each input and has measurable event evaluations.

Absolute continuity of a signed or complex measure with respect to a positive measure: $μ_{T} ≪ μ_{S}$ means every $μ_{S}$ -null measurable set is $μ_{T}$ -null.

Measurability of integration against a kernel: for a probability kernel and a nonnegative product-measurable function, the section integral is measurable.

Shalev-Shwartz and Ben-David, §3.2.1–3.2.2, PDF pp. 45–48: joint domain/label laws, marginal input law, classification risk, general loss, and expected risk.

Ben-David et al., §§3–4, printed pp. 154–158: source and target input distributions and labeling functions, their risks, and the joint-labeling error convention.

Mansour, Mohri, and Rostamizadeh, §§2–3, arXiv version 3, pp. 2–4: domain risks and distribution-dependent losses for arbitrary measurable function classes.

Sugiyama, §2, book pp. 928–930: training and test input laws, a shared conditional label law, and target generalization risk.

Depends on

Used by

Foundation models as broad-pretraining and adaptation systems Definition
H-delta-H divergence Definition
Loss discrepancy between domains Definition
A pretraining–adaptation interface with no shift guarantee Example
Calibration examples under validation and distribution shift Example
Covariate-shift weights can be unbounded and integrable Example
Identical input laws can hide a large joint-labeling error Example
Support mismatch prevents covariate-shift weighting Example
Covariate shift gives a joint-law density Lemma
Importance weighting under covariate shift Theorem
Loss-discrepancy domain-adaptation bound Theorem
Target risk bound by source risk, H-delta-H divergence, and joint error Theorem

Dependency tree · two levels

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Sources

Shai Shalev-Shwartz and Shai Ben-David, Understanding Machine Learning: From Theory to Algorithms, Chapters 2 and 3 (Cambridge University Press, 2014) (standard reference, not scraped)
Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan, A Theory of Learning from Different Domains, Machine Learning 79 (2010), §§3–4 (standard reference, not scraped)
Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh, Domain Adaptation: Learning Bounds and Algorithms, arXiv:0902.3430v3, §§2–3 (standard reference, not scraped)
Masashi Sugiyama, Learning under Non-stationarity: Covariate Shift Adaptation by Importance Weighting, Handbook of Computational Statistics, 2nd ed., Chapter 31 (2012) (standard reference, not scraped)