  Purpose: To develop a deep learning-based Bayesian inference for MRI
reconstruction. Methods: We modeled the MRI reconstruction problem with Bayes's
theorem, following the recently proposed PixelCNN++ method. The image
reconstruction from incomplete k-space measurement was obtained by maximizing
the posterior possibility. A generative network was utilized as the image
prior, which was computationally tractable, and the k-space data fidelity was
enforced by using an equality constraint. The stochastic backpropagation was
utilized to calculate the descent gradient in the process of maximum a
posterior, and a projected subgradient method was used to impose the equality
constraint. In contrast to the other deep learning reconstruction methods, the
proposed one used the likelihood of prior as the training loss and the
objective function in reconstruction to improve the image quality. Results: The
proposed method showed an improved performance in preserving image details and
reducing aliasing artifacts, compared with GRAPPA, $\ell_1$-ESPRiT, and MODL, a
state-of-the-art deep learning reconstruction method. The proposed method
generally achieved more than 5 dB peak signal-to-noise ratio improvement for
compressed sensing and parallel imaging reconstructions compared with the other
methods. Conclusion: The Bayesian inference significantly improved the
reconstruction performance, compared with the conventional $\ell_1$-sparsity
prior in compressed sensing reconstruction tasks. More importantly, the
proposed reconstruction framework can be generalized for most MRI
reconstruction scenarios.
