Bayesian Model Parameter Learning in Linear Inverse Problems: Application in EEG Focal Source Imaging

Inverse problems are often described as limited-data problems in which the signal of interest cannot be observed directly. Therefore, a physics-based forward model that relates the signal with the observations is typically needed. Unfortunately, unknown model parameters and imperfect forward models can still undermine the signal recovery. Even though supervised machine learning techniques offer promising avenues to improve the robustness of the solutions, we have to rely on model-based learning when there is no access to ground truth for the training. In this work, we studied a linear inverse problem that included an unknown non-linearly related model parameter and utilized a Bayesian model-based learning approach that allowed reliable signal recovery and subsequently estimation of the unknown model parameter. This approach, often referred to as Bayesian approximation error approach, employed a simplified model of the physics of the problem augmented with an approximation error term that compensated for the simplification. An error subspace was spanned with the help of the eigenvectors of the approximation error covariance matrix which allowed, alongside the primary signal, simultaneous estimation of the induced error. The estimated error and signal were then used to determine the unknown model parameter. For the model parameter estimation, we tested several different approaches: a conditional Gaussian regression, an iterative (model-based) optimization, and a Gaussian process that was modeled with the help of physics-informed learning. In addition, alternating optimization was used as a reference method. As an example application, we focused on the problem of reconstructing brain activity from EEG recordings (a.k.a. EEG source imaging) under the condition that the electrical conductivity of the patient’s skull was unknown in the model. Poorly selected conductivity values cause well-documented artifacts in the EEG source imaging results, and the determination of patient-specific head tissue conductivities is a significant technical problem. Our results demonstrated clear improvements in EEG source localization accuracy and provided feasible estimates for the unknown model parameter, skull conductivity.