Bayesian Inverse Problems and Kalman Filters

Abstract

We provide a brief introduction to Bayesian inverse problems and Bayesian estimators emphasizing their similarities and differences to the classical regularized least-squares approach to inverse problems. We then analyze Kalman filtering techniques for nonlinear systems, specifically the well-known Ensemble Kalman Filter (EnKF) and the recently proposed Polynomial Chaos Expansion Kalman Filter (PCE-KF), in this Bayesian framework and show how they relate to the solution of Bayesian inverse problems.