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 constant prior forecast

Example

Give $X \in {a, b}$ and $Y \in {1, 2, 3}$ their full power-set sigma-algebras. Let $X$ be uniform on ${a, b}$ and independent of $Y$ , whose probabilities are $(0.7, 0.2, 0.1)$ . The forecast $G (X) = (0.7, 0.2, 0.1)$ is fully reliable and equals the conditional class distribution given either feature value, but it is constant and therefore cannot distinguish $a$ from $b$ .

Example references

Probability calibration and calibration error: full reliability is defined by event identities on the forecast simplex.

Conditioning on a feature induces a calibrated forecast: a conditional class-probability forecast given any measurable feature is fully reliable.

Facts & Assumptions

Given: The independent finite-valued feature and label, and the constant forecast stated in the example.

[F1]

A forecast is fully reliable when each label-event probability conditional on forecast events equals the forecast coordinate integral. Probability calibration and calibration error

Verification

1.1givenalgebra

Independence gives $P (Y = j, X \in A) = P (Y = j) P (X \in A) = p_{j} P (X \in A)$ for every feature event $A$ and class $j$ . Thus the conditional label distribution given either $X = a$ or $X = b$ is $p = (0.7, 0.2, 0.1)$ , and $G$ is exactly that conditional forecast.

2.1F1step 1.1given∎

For any Borel set $B \subseteq Δ_{3}$ , if $p \in B$ then $P (Y = j, G \in B) = p_{j} = E [1_{{G \in B}} G_{j}]$ ; if $p \notin B$ , both sides are zero. By [F1], $G$ is fully reliable; its constancy means it has the same output at both feature values and so does not distinguish them.

Depends on

Probability calibration and calibration error

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)