Bayesian frequentist bounds for machine learning and system identification

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).