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.

A calibrated coarsened risk score loses information

Example

Give $X \in {a, b, c}$ and the binary label $Y \in {1, 2}$ their full power-set sigma-algebras. Let $X$ be uniform on ${a, b, c}$ and let the conditional probabilities be $P (Y = 1 ∣ X = a) = 0.1$ , $P (Y = 1 ∣ X = b) = 0.9$ , and $P (Y = 1 ∣ X = c) = 0.75$ . Coarsen the feature by $T (a) = T (b) = u$ and $T (c) = v$ , and forecast $G = (P (Y = 1 ∣ T), P (Y = 2 ∣ T))$ . Then $G (a) = G (b) = (0.5, 0.5)$ and $G (c) = (0.75, 0.25)$ . This forecast is fully reliable, although it cannot distinguish $a$ from $b$ , whose conditional label laws differ.

Example references

Probability calibration and calibration error: full reliability is the event-integral identity for every forecast-simplex event.

Conditioning on a feature induces a calibrated forecast: conditioning on a coarsened feature yields a fully reliable forecast.

Facts & Assumptions

Given: The finite feature-label law, coarsening map, and forecast stated in the example.

[F1]

Conditioning on any measurable feature gives a fully reliable forecast. Conditioning on a feature induces a calibrated forecast

[F2]

The calibration condition uses the forecast itself as the conditional probability vector given the forecast. Probability calibration and calibration error

Verification

1.1F1F2givenalgebra

The probability of $T = u$ is $2 / 3$ , and the conditional probability of $Y = 1$ there is $(0.1 + 0.9) / 2 = 0.5$ ; at $T = v$ it is $0.75$ . Thus the forecast is exactly the conditional label distribution given $T$ , and [F1] implies full reliability. Directly, each forecast value identifies $T$ , so the conditional distribution of $Y$ given $G = (0.5, 0.5)$ is $(0.5, 0.5)$ and given $G = (0.75, 0.25)$ is $(0.75, 0.25)$ , which is [F2].

2.1givenstep 1.1algebra∎

The original feature values $a$ and $b$ have different conditional label laws, with success probabilities $0.1$ and $0.9$ , but both map to the same forecast. Hence observing the forecast does not determine which of those feature values occurred, so the forecast discards information about $X$ while remaining calibrated.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Vaicenavicius et al., Evaluating Model Calibration in Classification, PMLR 89 (2019), Proposition 1 (standard reference, not scraped)