We explore a novel methodology for constructing confidence regions for parameters of linear models, using predictions from any arbitrary predictor. Our framework requires minimal assumptions on the noise and can be extended to functions deviating from strict linearity up to some adjustable threshold, thereby accommodating a comprehensive and pragmatically relevant set of functions. The derived confidence regions can be cast as constraints within a Mixed Integer Linear Programming framework, enabling optimisation of linear objectives. This representation enables robust optimization and the extraction of confidence intervals for specific parameter coordinates. Unlike previous methods, the confidence region can be empty, which can be used for hypothesis testing. Finally, we validate the empirical applicability of our method on synthetic data.

翻译：我们探索了一种构建线性模型参数置信域的新方法，该方法利用任意预测器的预测结果。该框架对噪声的假设要求极低，并可扩展至偏离严格线性关系但偏差不超过某个可调阈值的函数，从而涵盖了一类全面且具有实际相关性的函数。导出的置信域可表示为混合整数线性规划框架内的约束条件，从而能够优化线性目标函数。这种表示方式支持鲁棒优化，并可提取特定参数坐标的置信区间。与先前方法不同，该置信域可能为空，这可用于假设检验。最后，我们在合成数据上验证了该方法的实证适用性。