Optimization-Based Uncertainty Quantification for Ill-Posed Inverse Problems

Ill-posed inverse problems are situations where inferring the quantity of interest based on noisy data tends to produce extremely unstable solutions. It is customary to address this using regularization which reduces the variance of the estimates at the expense of an increased bias. While this can lead to well-behaved point estimates, it is challenging to provide rigorous frequentist uncertainties for the regularized estimates. In this talk, I will describe approaches that can be used to obtain improved frequentist uncertainty quantification in ill-posed problems.
The common theme of these methods is that they avoid explicit regularization by optimizing confidence bounds on functionals of the unknown quantity which implicitly regularizes the problem. This yields well-calibrated finite-sample frequentist uncertainties even in rank-deficient problems and in the presence of constraints. Special focus will be given to a new decision-theoretic approach for constructing such confidence bounds. Throughout this talk, I will demonstrate these ideas using case studies from particle physics and atmospheric remote sensing.