从共形预测分布中恢复预测密度

中文摘要 AI 辅助

共形预测区间用于在点预测周围构造边际校准良好的区域。近期工作侧重于将这些思想扩展到共形预测分布，其边际概率积分变换（PIT）服从Unif[0,1]分布。在本工作中，我们研究如何从共形预测分布中恢复预测密度。我们通过证明尾部修正版共形预测分布的渐近边际有效性，扩展了共形预测分布的理论。我们提出了一种称为分位数匹配的方法，该方法保持了边际PIT偏离均匀性的上界。此外，我们证明当分位数数量等于校准集大小时，分位数匹配诱导的分布等价于共形预测分布的清晰版本。对于共形密度的恢复，我们构建了一种保真度约束的核平滑带宽优化，该优化保持了渐近边际有效性并具有闭式解。我们在基于分层趋势模型的大型模拟房地产交易数据集上应用并比较了我们提出的方法。

英文摘要

Conformal prediction intervals are used to construct marginally well-calibrated regions around point predictions. Recent work has focused on extending these ideas to conformal predictive distributions, whose marginal probability integral transform (PIT) follows a Unif[0, 1] distribution. In this work, we investigate how to recover predictive densities from conformal predictive distributions. We extend the theory of conformal predictive distributions by proving asymptotic marginal validity for a tail-corrected version of conformal predictive distributions. We propose a method called quantile matching, which preserves an upper bound on the deviation of the marginal PIT from uniformity. Furthermore, we show that the distribution induced by quantile matching is equivalent to the crisp version of conformal predictive distributions when the number of quantiles equals the size of the calibration set. For the recovery of conformal densities, we construct a fidelity-constrained bandwidth optimization for kernel smoothing that preserves asymptotic marginal validity and has a closed-form solution. We apply and compare our proposed methodology on a large simulated real estate transactions dataset based on the Hierarchical Trend Model.

发表机构

• University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
• Delft University of Technology（代尔夫特理工大学）
• Ortec Finance（奥泰克金融）

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