ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Support mismatch prevents covariate-shift weighting

Example

Let $X = Y = {0, 1}$ with the discrete sigma-algebras. Let the source joint law put mass one on $(0, 0)$ and the target joint law put mass one on $(1, 0)$ . Both conditional label kernels assign label $0$ with probability one at each input. The input marginals are $μ_{S} = δ_{0}$ and $μ_{T} = δ_{1}$ . Although the conditional label laws agree, there is no nonnegative measurable $w$ satisfying $μ_{T} (A) = \int_{A} w d μ_{S}$ for every measurable $A$ .

Example references

Source and target joint laws, risks, and covariate shift: defines the joint laws, conditional label kernels, input marginals, and the covariate-shift condition.

Importance weighting under covariate shift: requires absolute continuity of the target input marginal with respect to the source marginal.

Ben-David et al., §3, printed pp. 154–155: distinguishes shifts in input laws from changes in labeling functions.

Facts & Assumptions

Given: The two finite joint laws and their common constant conditional label kernel in the Example.

[F1]

Source and target joint laws, risks, and covariate shift: covariate shift requires $μ_{T} ≪ μ_{S}$ as well as equality of conditional label kernels $μ_{T}$ -almost everywhere.

[F2]

A nonnegative integral over a null set vanishes: the integral of a nonnegative measurable function over a measurable null set is zero.

Verification

1.1F1given

For $A = {1}$ , the specified point masses give $μ_{S} (A) = 0$ and $μ_{T} (A) = 1$ . Thus $μ_{T} ≪̸ μ_{S}$ by the definition of absolute continuity.

2.1F2step 1.1algebra

If a nonnegative measurable density $w$ satisfied the proposed setwise identity, then $\int_{A} w d μ_{S} = 0$ by [F2], contradicting $μ_{T} (A) = 1$ from step 1.1.

3.1F1step 1.1∎

The kernels agree everywhere, but the absolute-continuity hypothesis in [F1] fails by step 1.1, so the covariate-shift importance-weighting theorem does not apply.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Shai Ben-David et al., A Theory of Learning from Different Domains, Machine Learning 79 (2010), §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), §3.1 (standard reference, not scraped)