We revisit the problem of constructing predictive confidence sets for which we wish to obtain some type of conditional validity. We provide new arguments showing how ``split conformal'' methods achieve near desired coverage levels with high probability, a guarantee conditional on the validation data rather than marginal over it. In addition, we directly consider (approximate) conditional coverage, where, e.g., conditional on a covariate $X$ belonging to some group of interest, we would like a guarantee that a predictive set covers the true outcome $Y$. We show that the natural method of performing quantile regression on a held-out (validation) dataset yields minimax optimal guarantees of coverage here. Complementing these positive results, we also provide experimental evidence that interesting work remains to be done to develop computationally efficient but valid predictive inference methods.

翻译：我们重新审视构建预测置信集的问题，旨在获得某种类型的条件有效性。我们提出了新的论证，表明"拆分共形"方法如何以高概率实现接近期望的覆盖水平，这一保证条件于验证数据而非对其取边际。此外，我们直接考虑（近似）条件覆盖问题，例如在给定协变量$X$属于某个感兴趣群体的条件下，我们希望预测集能覆盖真实结果$Y$。我们证明，在留出（验证）数据集上执行分位数回归的自然方法能够在此处获得极小极大最优的覆盖保证。与这些积极结果相补充，我们还通过实验证据表明，要开发计算高效且有效的预测推断方法，仍有大量有意义的工作有待完成。