  End-to-End (E2E) unrolled optimization frameworks show promise for Magnetic
Resonance (MR) image recovery, but suffer from high memory usage during
training. In addition, these deterministic approaches do not offer
opportunities for sampling from the posterior distribution. In this paper, we
introduce a memory-efficient approach for E2E learning of the posterior
distribution. We represent this distribution as the combination of a
data-consistency-induced likelihood term and an energy model for the prior,
parameterized by a Convolutional Neural Network (CNN). The CNN weights are
learned from training data in an E2E fashion using maximum likelihood
optimization. The learned model enables the recovery of images from
undersampled measurements using the Maximum A Posteriori (MAP) optimization. In
addition, the posterior model can be sampled to derive uncertainty maps about
the reconstruction. Experiments on parallel MR image reconstruction show that
our approach performs comparable to the memory-intensive E2E unrolled
algorithm, performs better than its memory-efficient counterpart, and can
provide uncertainty maps. Our framework paves the way towards MR image
reconstruction in 3D and higher dimensions
