Papers

Working papers and technical notes from the site maintainer

Marginally Useful

Formalizing the Information Gap in Conformal Prediction

Conformal prediction gives a distribution-free, finite-sample guarantee of marginal coverage for a set, and it is easy to read this as more than it is. The paper separates coverage from forecast quality: for a fixed location predictor re-leveled by a single residual shape, the log-score regret to the oracle is exactly the mutual information between the residual and the input — the quantity conformalization cannot touch.

A Feynman–Wigner Diagnostic for Conformal Prediction

via signed de Finetti representations

Conformal’s guarantee is exact under exchangeability, and exchangeability is what de Finetti’s theorem describes. In the finite form the mixing measure may be signed (Kerns–Székely) — a Feynman–Wigner negative probability. The sign of that measure decides whether conformal’s per-case coverage stays adaptive or fans: ordinary data is fine, while ranked, compositional, or contest scores sit in the signed corner. The marginal number never moves.

Betting Against a Conformal Predictor

a parimutuel account of the information gap

The same gap, derived from a betting mechanism rather than a score. Treat the predictor as the crowd in a parimutuel pool on the residual; in the continuous limit the payoff is the ratio of your density to the crowd’s. An entrant who knows only the marginal breaks even; that is marginal coverage as wealth. One who conditions on the input grows his bankroll at rate exactly the mutual information. The gap is the rent, and the mechanism is the one the microprediction platform and the MidOne contest actually ran.

The Width of the Conformal Fan

dependence and the variance of realized coverage

Fix the calibration set and the coverage you realize is a random number around 1−α — the Beta “fan.” Its width is set by the sign of the cross-sample dependence: positive dependence adds a non-negative between-dataset term (exact, via de Finetti and the law of total variance) and empirically widens it; negative dependence narrows it (proved in aggregate, conjectured per level), to exactly zero at the contest floor, where the realized coverage equals k/(n+1) every time. The same sign the signed-de-Finetti note attaches to within-dataset adaptivity, now governing across-dataset spread.

A Contragredient View of Conformal Placement and Steinitz Balancing

the exact orbit identity, its balanced-permutation primal, herding as the relaxation — rough draft, not for dissemination

Conformal validity stated as an exact orbit-averaging identity for the permutation action, in contragredient form through the Reynolds projector. The primal construction under the same action is Steinitz balancing; conformal placements are themselves a zero-sum population, and an explicit balanced ordering keeps the running placement-acceptance average within 1/(2t) of the orbit level k/(n+1), exact on the full orbit. The measure-side counterpart of calibration on the function–measure pairing is balanced sampling and exact cubature. Herding is the with-replacement relaxation, confined to a remark.

The Two Prices of Dependence

a vanishing coverage tax and a fixed sharpness rent

Temporal dependence charges split conformal twice, on different scales. The coverage tax vanishes with the calibration size (Barber–Pananjady: at most min_τ{τ/(n+1)+2β(τ)}). The sharpness rent does not: the log-score regret of the marginal predictive against the past-conditional oracle equals the entropy-rate gap ΔH = I(S₀; past), the information-gap identity with the past as the side information. For Gaussian scores it exponentiates into width: oracle intervals are e^−ΔH narrower at matched coverage, √(1−φ²) for AR(1). For the certificate, dependence is a rounding error; for the forecast, it is the whole opportunity.