Technical note

The width of the conformal fan

Dependence and the variance of realized coverage. A companion note to the Feynman–Wigner diagnostic.

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The idea

Fix the calibration set and the coverage you then realize is a random number, not \(1-\alpha\) exactly. For independent scores it follows the Beta\((k,n-k+1)\) law — the “fan,” with variance about \(\alpha(1-\alpha)/n\). This note shows the width of that fan is set by the sign of the cross-sample dependence. The mean stays pinned at the nominal level; the variance, to leading order, is the average pairwise covariance of the exceedance indicators at the operating quantile, times the independent fan.

What the sign does

Positive (extendable) dependence adds a between-dataset term and widens the fan — proved exactly through de Finetti and the law of total variance. Negative dependence narrows it, to exactly zero at the maximally negatively associated contest floor \(\rho=-1/(n-1)\), where the realized coverage equals the nominal level on every draw. It is the same sign the companion note reads off the finite de Finetti measure: a genuine prior widens the fan, the signed corner collapses it.

What is proved, and what is left

Exact: the de Finetti decomposition for positive dependence, and the zero at the contest floor. Asymptotic, with an explicit reduction factor: the fan coefficient in general. The remaining finite-sample inequality — negative association never widens the fan — rests on a convex-order contraction of the exceedance count (proved) plus a single-crossing step, and is confirmed numerically across dependence structures by check_fan.py. The stronger dispersive ordering is deliberately not claimed; it can fail at fixed marginals. A constructive corollary: a balanced, transductively symmetric calibration set tightens coverage without disturbing the marginal guarantee.

Using conformal prediction in your own project? Tell Claude: “Read https://conformalprediction.net/SKILL.md and create a project skill from it.” It adds a check for whether your coverage is conditionally trustworthy.