TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Importance weighting under covariate shift

Statement

Assume AC. Let $P_{S}, P_{T}$ be source and target joint laws under covariate shift, with input marginals $μ_{S}, μ_{T}$ and common conditional label kernel $K$ . Let $w$ be a nonnegative measurable representative of $d μ_{T} / d μ_{S}$ . For any measurable predictor $h$ and nonnegative product-measurable loss $ℓ$ , if the weighted source loss is integrable, then $R_{T} (h) = \int_{X \times Y} ℓ (h (x), y) P_{T} (d (x, y)) = \int_{X \times Y} w (x) ℓ (h (x), y) P_{S} (d (x, y)) < + \infty .$

Statement references

Sugiyama, §3.1, book pp. 930–931: the importance-weighting identity for a shared conditional label law.

Mansour, Mohri, and Rostamizadeh, §2, arXiv version 3, pp. 2–3: domain risks and arbitrary measurable loss notation.

Facts & Assumptions

Given: AC; source and target joint laws under covariate shift; a measurable predictor $h$ ; a nonnegative product-measurable loss $ℓ$ ; and $\int w (x) ℓ (h (x), y) P_{S} (d (x, y)) < + \infty$ .

[A1]

The Axiom of Choice: AC is assumed for the Radon-Nikodym representatives used by the likelihood-density and change-of-measure suppliers.

[F1]

Source and target joint laws, risks, and covariate shift: under covariate shift, $μ_{T} ≪ μ_{S}$ and the source and target conditional label kernels agree $μ_{T}$ -almost everywhere.

[F2]

Likelihood densities: in the two-law experiment dominated by $μ_{S}$ , AC supplies a nonnegative measurable representative $w$ of $d μ_{T} / d μ_{S}$ .

[F3]

Covariate shift gives a joint-law density: $P_{T} ≪ P_{S}$ , with nonnegative likelihood-ratio representative $r (x, y) = w (x)$ on the joint space.

[F4]

Population and empirical risks are measurable: in a supervised prediction problem, a measurable predictor in the predictor class induces a product-measurable loss function.

[F5]

Likelihood-ratio change of measure: under AC and absolute continuity, $\int g d P_{T} = \int g r d P_{S}$ for every nonnegative measurable $g$ .

Proof

1.1F4given

Set $g_{h} (x, y) : = ℓ (h (x), y)$ . For each $D \in {S, T}$ , the given measurable spaces, joint law $P_{D}$ , measurable predictor $h$ , and product-measurable loss $ℓ$ form a supervised prediction problem with predictor class ${h}$ . Since $h \in {h}$ , the hypotheses of [F4] hold and $g_{h}$ is product-measurable. It is nonnegative, so both integrals in the claim are defined as extended nonnegative integrals.

2.1A1F1F2F3F5givenalgebra∎

By [F1] and [F2], the covariate-shift hypotheses hold and $w$ is an available input-marginal density. The joint-law lemma [F3] identifies the joint density as $r (x, y) = w (x)$ . Apply [F5] to $g_{h}$ to obtain $\int g_{h} d P_{T} = \int g_{h} r d P_{S} = \int w (x) ℓ (h (x), y) d P_{S}$ . The assumed integrability makes this common value finite, proving the stated identity.

Depends on

Used by

Dependency tree · two levels

35 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

Masashi Sugiyama, Learning under Non-stationarity: Covariate Shift Adaptation by Importance Weighting, Handbook of Computational Statistics, 2nd ed., Chapter 31 (2012), §3.1 (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)