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Marginal risk control, calibration, and adaptive reuse

Scope and limitations

The finite-grid theorem controls expected test loss after averaging over both the calibration rows and the test row. It is not a high-probability guarantee on the conditional risk after the realized calibration sample. For a two-state illustration, let $U \in {0, 1}$ represent the realized calibration state with $P (U = 0) = P (U = 1) = 1 / 2$ , let $T = σ (U)$ , and let the test loss $X$ equal $0$ when $U = 0$ and $1 / 2$ when $U = 1$ . Then the conditional risk $R = E [X ∣ T]$ equals $X$ , so $E [R] = 1 / 4$ while $P (R > 1 / 4) = 1 / 2$ . Thus expected-risk control alone does not imply a high-probability bound on conditional risk. The finite-grid theorem and its monotone-envelope extension give no general risk bound conditional on an arbitrary subgroup event or covariates. They do give the corresponding conditional bound when the full loss matrix has an actual regular conditional law given the sigma-algebra generated by that information, its sections are exchangeable, and the applicable boundedness, nesting, and endpoint-safety assumptions hold almost surely under those sections. The results give no validity when the deployment distribution differs from the exchangeable calibration and test distribution or when arbitrary reuse of calibration rows breaks exchangeability. For a classifier on a random labeled input, let $Y$ and $\hat{Y}$ be its true and predicted labels and let $C \in [0, 1]$ be its measurable reported confidence. Top-label probability calibration means $E [1 {Y = \hat{Y}} ∣ σ (C)] = C$ almost surely; this is a different claim and is not concluded here. Split conformal coverage controls the particular miscoverage event under its own exchangeability hypotheses; it is not a guarantee for every bounded loss. For nonmonotone losses, the finite tail-maximum theorem applies but can increase the loss and make the selected decision more conservative. The conditional form proved here for an independently trained model assumes an actual regular conditional exchangeable law of the calibration and test rows, with the stated hypotheses holding almost surely.

Scope and limitations references

Finite-sample expected-risk control on a finite grid: expected test loss is averaged over calibration and test rows.

Finite-grid risk control by monotone envelopes: finite tail maxima dominate the original losses pointwise.

Split conformal marginal coverage: the exchanged split-score rank theorem controls marginal miscoverage under its stated conditional exchangeability hypotheses.

Regular conditional distribution: conditional distribution means a probability kernel satisfying the conditional-event identity.

Angelopoulos et al., Conformal Risk Control, §§1.2, 2.3–2.4, 4.1

Angelopoulos, Barber, and Bates, Theoretical Foundations of Conformal Prediction, Chapter 4

Shafer and Vovk, A Tutorial on Conformal Prediction, §§2.2, 3.1–3.2, 4.2–4.5

Facts & Assumptions

Given: The expected-risk theorem, the split-conformal marginal-coverage theorem, and the finite two-state conditional-risk illustration in the Statement.

[F1]

Under exchangeability, boundedness, loss monotonicity, and endpoint safety, the finite-grid theorem bounds expected test loss averaged over calibration and test rows. It also gives a conditional bound when an actual regular conditional law given $T$ has exchangeable sections satisfying those hypotheses almost surely. Finite-sample expected-risk control on a finite grid

[F2]

Split conformal controls the marginal coverage probability of its particular set-valued predictor under its stated conditional exchangeability and score hypotheses. Split conformal marginal coverage

[F3]

The finite tail-maximum theorem bounds the original loss pointwise by a finite monotone envelope. Finite-grid risk control by monotone envelopes

[F4]

A regular conditional distribution is a probability kernel satisfying the conditional-event identity for every conditioning event and measurable target set. Regular conditional distribution

[F5]

A conditional-expectation version is measurable with respect to the conditioning sigma-algebra and has the same integrals as the original variable on every event in that sigma-algebra. Conditional expectation given a sigma algebra

[F6]

The expectation of a bounded nonnegative random variable is its Lebesgue integral against the probability measure. Expectation of a nonnegative or integrable random variable

Verification

1.1givenF5F6algebra

The variable $X$ is $T$ -measurable and integrable, so it is itself a version of $E [X ∣ T]$ by [F5]. Therefore $R = X$ , $E R = (1 / 2) 0 + (1 / 2) (1 / 2) = 1 / 4$ , and $P (R > 1 / 4) = 1 / 2$ . In particular, the chance constraint $P (R \leq 1 / 4) \geq 1 - δ$ fails for every $δ < 1 / 2$ .

1.2F1F2F4

[F1] gives a general marginal bound and a conditional bound only when its actual regular conditional law is sectionwise exchangeable and boundedness, loss nesting, and endpoint safety hold almost surely under those sections. Thus neither [F1] nor [F2] gives a general guarantee conditional on an arbitrary subgroup or feature value; [F2] concerns a particular coverage event under its own conditional-exchangeability hypotheses. An adaptive reuse or distribution shift that violates the applicable exchangeability hypotheses falls outside [F1] and [F2].

2.1F1F3F4∎

The conditional extension in [F1] requires the actual regular conditional law described by [F4]. The envelope in [F3] only dominates the original loss pointwise, so it may increase the selected decision's risk and make the resulting bound more conservative.

Depends on

Used by

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Sources

Anastasios N. Angelopoulos, Stephen Bates, Adam Fisch, Lihua Lei, and Tal Schuster, Conformal Risk Control, arXiv:2208.02814v4 (standard reference, not scraped)
Anastasios N. Angelopoulos, Rina Foygel Barber, and Stephen Bates, Theoretical Foundations of Conformal Prediction, prepublication book manuscript (standard reference, not scraped)
Glenn Shafer and Vladimir Vovk, A Tutorial on Conformal Prediction, Journal of Machine Learning Research 9 (2008), 371–421 (standard reference, not scraped)