LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pass

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.

Class-conditional mean risk scores are areas under threshold rates

Statement

Fix a group $t \in {1, 2}$ with a finite joint law $\Pr_{t}$ for a binary outcome $Y \in {0, 1}$ and a score $S$ taking finitely many values in $[0, 1]$ , and assume $\Pr_{t} (Y = 1) > 0$ and $\Pr_{t} (Y = 0) > 0$ . For each threshold $u \in [0, 1]$ , define the true-positive rate ${TPR}_{t} (u) = \Pr_{t} (S \geq u ∣ Y = 1)$ and the false-positive rate ${FPR}_{t} (u) = \Pr_{t} (S \geq u ∣ Y = 0)$ . These are finite step functions and

E_{t} [S ∣ Y = 1] = \int_{0}^{1} {TPR}_{t} (u) d u, E_{t} [S ∣ Y = 0] = \int_{0}^{1} {FPR}_{t} (u) d u .

The integrals are ordinary Riemann integrals of the finite step functions.

Statement references

KMR's finite group-conditioned risk-score assignment and calibration: provides the finite group-conditional law and the finite $[0, 1]$ -valued score used in this statement.

Kleinberg, Mullainathan, and Raghavan, complete 23-page paper: §§1.1–2 define groupwise positive/negative score means and use these class-conditioned score distributions in Theorem 1.1.

Facts & Assumptions

Given: A group $t \in {1, 2}$ in the finite-bin risk-score model. Write its distinct score values as $s_{1}, \dots, s_{m}$ ; conditional on $Y = 1$ let their probabilities be $w_{j}^{+}$ , and conditional on $Y = 0$ let them be $w_{j}^{-}$ . Prove that each class-conditional mean score is the area under its corresponding threshold rate.

[F1]

KMR's finite group-conditioned risk-score assignment and calibration: the feature, outcome, bin, and score variables have a finite joint law in each group, and each score value lies in $[0, 1]$ .

Proof

1.1givenF1algebra

For each $j$ , the threshold indicator is $1_{[0, s_{j}]} (u)$ . Hence ${TPR}_{t} (u) = \sum_{j = 1}^{m} w_{j}^{+} 1_{[0, s_{j}]} (u)$ . Integrating this finite sum term by term gives $\int_{0}^{1} {TPR}_{t} (u) d u = \sum_{j = 1}^{m} w_{j}^{+} s_{j} = E_{t} [S ∣ Y = 1]$ .

2.1givenF1algebra∎

The same indicator calculation gives ${FPR}_{t} (u) = \sum_{j = 1}^{m} w_{j}^{-} 1_{[0, s_{j}]} (u)$ and $\int_{0}^{1} {FPR}_{t} (u) d u = \sum_{j = 1}^{m} w_{j}^{-} s_{j} = E_{t} [S ∣ Y = 0]$ .

Remarks

The identity uses only finite support; no interchange theorem or choice principle is needed.

Depends on

KMR's finite group-conditioned risk-score assignment and calibration

Used by

Dependency tree · two levels

2 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

Jon Kleinberg, Sendhil Mullainathan, and Manish Raghavan, Inherent Trade-Offs in the Fair Determination of Risk Scores, arXiv:1609.05807v2 (2016), §§1.1–2 (standard reference, not scraped)
Applied Statistics: From Data to Decisions, MS&E 125, Fairness in Algorithmic Decision-Making, section ‘The impossibility: you cannot satisfy them all’ (standard reference, not scraped)