协变量偏移下共形预测的锐利训练条件覆盖

中文摘要 AI 辅助

加权分裂共形预测通过测试与训练协变量分布之间的似然比对校准分数进行重新加权，并在协变量偏移下保证边际覆盖。我们研究其覆盖在给定校准数据条件下的性质。一个基于固定总体分位数处单一集中不等式的初等论证，给出了无未指定常数的显式训练条件界，并表明相关尺度并非似然比的上确界，而是由偏移的卡方散度和测试总体中概率等于错误覆盖水平、似然比最大的那部分上的比值平均值构建的方差代理。一个两点下界表明，根号m速率和卡方贡献是偏移所固有的。在显式膨胀的水平下运行，加权分位数成为一个确定性的PAC预测集。我们将其与随机拒绝采样和重要性加权先测试后学习进行比较，并通过认证选择似然比的裁剪水平，描绘了每种方法给出更窄有效集的区域。该分析扩展到估计的似然比和从未标记源样本估计的尾部泛函，从而产生一个完全有限样本的证书。

英文摘要

Weighted split conformal prediction reweights calibration scores by the likelihood ratio between the test and training covariate distributions and guarantees marginal coverage under covariate shift. We study its coverage conditional on the calibration data. An elementary argument, based on a single concentration inequality at a fixed population quantile, gives explicit training-conditional bounds without unspecified constants, and shows that the relevant scale is not the supremum of the likelihood ratio but a variance proxy built from the chi-squared divergence of the shift and from the average of the ratio over the part of the test population, of probability equal to the miscoverage level, where it is largest. A two-point lower bound shows that the root-m rate and the chi-squared contribution are intrinsic to the shift. Run at an explicitly inflated level, the weighted quantile becomes a deterministic PAC prediction set. We compare it with randomized rejection sampling and with importance-weighted learn-then-test and, through a certified choice of a clipping level for the likelihood ratio, map the regime in which each gives the narrower valid set. The analysis extends to estimated likelihood ratios and to tail functionals estimated from an unlabeled source sample, which yields a fully finite-sample certificate.

发表机构

• Mofid Securities（莫菲德证券公司）

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