用β分布建模校准后覆盖率，量化非独立同分布偏差。
Conformal Prediction via Transported Beta Laws
- 以β分布为参考，通过沃尔什距离度量覆盖率偏离程度。
- 在依赖数据下仍能保证覆盖率与坏校准概率的严格界限。
- 适用于数据偏移、聚类、平稳混合等场景，可直接计算偏差。
分拆式共形预测在可交换性假设下提供有限样本的边际覆盖保证，但该保证是对随机校准样本的平均。本文研究由实际共形阈值诱导的校准条件覆盖律。在连续独立同分布情形下，该律精确服从Beta(k, n+1−k)分布，通常的边际保证即为其均值。本文将此β分布作为有限样本参照对象，利用[0,1]上沃尔什距离量化其偏离程度。该框架直接给出边际覆盖差距和坏校准概率的界，并根据变形方式区分不同非独立同分布行为：测试侧偏移通过覆盖尺度上的传输映射作用，而校准依赖则改变顺序统计量本身的分布。在尺度偏移、聚类及平稳混合设定中，可显式或通过Berry-Esseen近似刻画变形。对依赖过程的模拟验证了首阶近似能有效追踪经验沃尔什距离，即使在中等样本量下亦成立。
原文摘要 · Abstract (English)
Split conformal prediction provides finite-sample marginal coverage under exchangeability, but this guarantee averages over the random calibration sample. We study instead the law of the calibration-conditional coverage induced by a realized conformal threshold. In the continuous i.i.d. setting this law is exactly $Beta(k,n+1-k)$, so the usual marginal guarantee corresponds to its mean. We take this beta law as a finite-sample reference object and quantify departures from it using Wasserstein distances on $[0,1]$. The framework yields direct bounds on marginal coverage gaps and on bad-calibration probabilities, and separates different sources of non-i.i.d. behavior according to how they deform the beta reference: test-side shift acts through a transport map on the coverage scale, while calibration dependence changes the order-statistic law itself. We instantiate the framework in scale-shift, clustered, and stationary mixing settings, where the induced deformations can be characterized explicitly or through Berry-Esseen approximations. Simulations on dependent processes confirm that the first-order approximation tracks the empirical Wasserstein distance even at moderate sample sizes.
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