The ability to reconstruct high-quality images from undersampled MRI data is vital in improving MRI temporal resolution and reducing acquisition times. Deep learning methods have been proposed for this task, but the lack of verified methods to quantify the uncertainty in the reconstructed images hampered clinical applicability. We introduce "NPB-REC", a non-parametric fully Bayesian framework, for MRI reconstruction from undersampled data with uncertainty estimation. We use Stochastic Gradient Langevin Dynamics during training to characterize the posterior distribution of the network parameters. This enables us to both improve the quality of the reconstructed images and quantify the uncertainty in the reconstructed images. We demonstrate the efficacy of our approach on a multi-coil MRI dataset from the fastMRI challenge and compare it to the baseline End-to-End Variational Network (E2E-VarNet). Our approach outperforms the baseline in terms of reconstruction accuracy by means of PSNR and SSIM (34.55, 0.908 vs. 33.08, 0.897, p < 0.01, acceleration rate R = 8) and provides uncertainty measures that correlate better with the reconstruction error (Pearson correlation, R = 0.94 vs. R = 0.91). Additionally, our approach exhibits better generalization capabilities against anatomical distribution shifts (PSNR and SSIM of 32.38, 0.849 vs. 31.63, 0.836, p < 0.01, training on brain data, inference on knee data, acceleration rate R = 8). NPB-REC has the potential to facilitate the safe utilization of deep learning-based methods for MRI reconstruction from undersampled data. Code and trained models are available at https://github.com/samahkh/NPB-REC.
