CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Conditioning on a feature induces a calibrated forecast

Statement

Assume AC. Let a finite-label supervised prediction problem have measurable instance space $(X, Σ_{X})$ and finite labels $Y = {y_{1}, \dots, y_{K}}$ with the full power-set sigma-algebra, where $K \geq 2$ . Identify label $y_{j}$ with its index $j$ when applying numbered-label calibration, so forecast coordinate $j$ always represents the event $Y = y_{j}$ . For every measurable feature $T : X \to (Z, Σ_{Z})$ , there is a measurable $η_{T} : Z \to Δ_{K}$ such that $G = η_{T} (T (X))$ is a conditional class-probability forecast given $T (X)$ and is fully reliable. This includes the identity feature and a constant feature.

Statement references

A finite label has a conditional probability function: supplies the measurable simplex-valued conditional probability function and its event identities under AC.

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

Facts & Assumptions

Given: AC, a finite-label supervised prediction problem with coordinate random variables $X, Y$ , and a measurable feature $T : X \to (Z, Σ_{Z})$ .

[A1]

Assume AC, as used in the finite-label conditional-probability lemma. The Axiom of Choice

[F1]

There is a measurable $η : X \to Δ_{K}$ such that $P (X \in A, Y = y_{j}) = E [η_{j} (X) 1_{{X \in A}}]$ for every measurable $A$ and label $j$ . A finite label has a conditional probability function

[F2]

Full reliability means $P (Y = y_{j}, G \in B) = E [1_{{G \in B}} G_{j}]$ for every Borel $B \subseteq Δ_{K}$ and each label $j$ . Probability calibration and calibration error

[F3]

A measurable map has measurable preimages of measurable sets. A measurable function between measurable spaces

Proof

1.1A1F1F2F3given

Apply [F1] to the joint law of $(T (X), Y)$ on $Z \times Y$ , which is a measurable joint probability law because $T$ is measurable. Equip this law with simplex-valued constant predictors and the zero loss to obtain a supervised prediction problem; [F1] only uses its joint law. It supplies a measurable $η_{T} : Z \to Δ_{K}$ with $P (T (X) \in A, Y = y_{j}) = E [η_{T, j} (T (X)) 1_{{T (X) \in A}}]$ for every $A \in Σ_{Z}$ . Set $G = η_{T} (T (X))$ . For a Borel set $B \subseteq Δ_{K}$ , [F3] makes $A = η_{T}^{- 1} (B)$ measurable; the event identity then gives $P (Y = y_{j}, G \in B) = E [G_{j} 1_{{G \in B}}]$ . This is precisely the reliability identity [F2].

2.1step 1.1∎

Since $j$ and $B$ were arbitrary, all label and Borel-event identities hold, so $G$ is fully reliable.

Source notes

Vaicenavicius et al. Proposition 1 states the reliability of a conditional class-probability forecast. The event-integral proof above makes the version and measurable-event hypotheses explicit.

Depends on

Used by

A calibrated coarsened risk score loses information Example

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Sources

Vaicenavicius et al., Evaluating Model Calibration in Classification, PMLR 89 (2019), Proposition 1 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, 5th ed., Section 4.1 (standard reference, not scraped)