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.

Covariate-shift weights can be unbounded and integrable

Example

Let $X = N_{\geq 1}$ with the discrete sigma-algebra and $Y = {0, 1}$ . Let the source input law have masses $μ_{S} ({n}) = 2^{- n}$ and the target input law have masses $μ_{T} ({n}) = n 2^{- (n + 1)}$ for $n \geq 1$ . In both domains the label is $0$ with probability one. Use the zero-one loss $ℓ (a, y) = 1_{{a \neq y}}$ and the constant classifier $h (n) = 1$ . The covariate-shift weight is $w (n) = n / 2$ : it is unbounded, but the weighted loss is integrable and the target risk equals the importance-weighted source risk, both equal to $1$ . Thus the exact risk identity holds while the bounded-weight hypothesis in any finite-sample concentration result that requires it fails.

Example references

Source and target joint laws, risks, and covariate shift: defines source and target joint laws, marginal laws, risks, and covariate shift.

Importance weighting under covariate shift: establishes the risk identity when the weighted loss is integrable.

Sugiyama, §3.1, book pp. 930–931: importance-weighted risk under covariate shift and the distinction between the exact identity and estimation methods.

Facts & Assumptions

Given: The discrete source and target masses, their common deterministic label kernel, the constant classifier $h = 1$ , and zero-one loss.

[F1]

Source and target joint laws, risks, and covariate shift: input marginals are probability measures, and under covariate shift their conditional label kernels agree and $μ_{T} ≪ μ_{S}$ .

[F2]

Importance weighting under covariate shift: for an integrable weighted loss, target risk is its target-input-density-weighted source risk.

Verification

1.1givenconstructalgebra

For $N \geq 1$ , $\sum_{n = 1}^{N} 2^{- n} = 1 - 2^{- N}$ , so the source masses sum to $1$ . Also $\sum_{n = 1}^{N} n 2^{- n} = 2 - (N + 2) 2^{- N}$ : this holds at $N = 1$ , and adding $(N + 1) 2^{- (N + 1)}$ proves the formula at $N + 1$ . Since $N 2^{- N} \to 0$ , the target masses sum to $\frac{1}{2} \sum_{n \geq 1} n 2^{- n} = 1$ .

2.1F1givenstep 1.1algebra

Every source atom has positive mass, so the only $μ_{S}$ -null set is empty and $μ_{T} ≪ μ_{S}$ . The atomwise ratios satisfy $μ_{T} ({n}) / μ_{S} ({n}) = n / 2$ , hence $w (n) = n / 2$ is a density and $\sup_{n} w (n) = + \infty$ . The two conditional label kernels are identical.

3.1F2givenstep 1.1step 2.1∎

The loss of $h (n) = 1$ against label $0$ is $1$ , hence $R_{T} (h) = 1$ . Its weighted source risk is $\sum_{n \geq 1} 2^{- n} (n / 2) = 1$ by step 1.1; thus it is finite and equals the target risk.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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), §§2–3.1 (standard reference, not scraped)