Estimating a function from noisy measurements is a crucial problem in statistics and engineering, with an impact on machine learning predictions and identification of dynamical systems. In view of robust control design and safety-critical applications such as autonomous driving and smart healthcare, estimates are required to be complemented with uncertainty bounds quantifying their reliability. Most of the available results are derived by constraining the estimates to belong to a deterministic function space; however, the returned bounds often result overly conservative and, hence, of limited usefulness. An alternative is to use a Bayesian framework. The regions thereby obtained however require complete specification of prior distributions whose choice may significantly affect the probability of inclusion. This study presents a framework for the effective computation of regions that include the unknown function with exact probability. In this setting, the users not only have the freedom to modulate the amount of prior knowledge that informs the constructed regions but can, on a different plane, finely modulate their commitment to such information. The result is a versatile certified estimation framework capable of addressing a multitude of problems, ranging from parametric estimation (where the probabilistic guarantees can be issued under no commitment to the prior information) to non-parametric problems (that call for fine exploitation of prior information).

从噪声测量中估计函数是统计学和工程中的一个关键问题，对机器学习预测和动态系统的识别有影响。鉴于稳健的控制设计和安全关键型应用（例如自动驾驶和智能医疗保健），估计需要辅以量化其可靠性的不确定性界限。大多数可用结果是通过将估计限制为属于确定性函数空间而得出的；然而，返回的边界通常会导致过于保守，因此用处有限。另一种方法是使用贝叶斯框架. 然而，由此获得的区域需要完整说明先验分布，其选择可能会显着影响包含的概率。本研究提出了一个有效计算区域的框架，其中包括具有精确概率的未知函数。在这种情况下，用户不仅可以自由调整通知构建区域的先验知识量，而且可以在不同的平面上精细调整他们对此类信息的承诺。结果是一个通用的认证估计框架，能够解决多种问题，从参数估计（在不承诺先验信息的情况下可以发布概率保证）到非参数问题（需要精细利用先验信息） ）。