Paper:
Bayesian inference is used extensively to quantify the uncertainty in an inferred field given the measurement of a related field when the two are linked by a mathematical model. Despite its many applications, Bayesian inference faces challenges when inferring fields that have discrete representations of large dimension, and/or have prior distributions that are difficult to characterize mathematically. In this work we demonstrate how the approximate distribution learned by a generative adversarial network (GAN) may be used as a prior in a Bayesian update to address both these challenges. We demonstrate the efficacy of this approach by inferring and quantifying uncertainty in inference problems arising in computer vision and physics-based applications. In both instances we highlight the role of computing uncertainty in providing a measure of confidence in the solution, and in designing successive measurements to improve this confidence.

Review:
Summary of the paper:
  
        The paper proposes a Bayesian approach to make inference about latent variables such as un-corrupted images. The prior distribution plays a key role in this task. The authors use a GAN to estimate this prior distribution. Then, standard Bayesian techniques such a Hamilton Monte Carlo are used to make inference about the latent variables.
Detailed comments:
Eq. (8) is expected to give very bad results. The reason is that it is very unlikely to sample from the prior configurations for z that are compatible with y.
The paper does not address learning any model parameters. e.g. the amount of noise.
A more principled approach would be to estimate the prior parameters using maximum likelihood estimation. That has already been done in the case of the
variational autoencoder.
The variational autoencoder is an already known method that can be used to solve the problem formulated by the authors. It also automatically proposes
an inference network that can be used for recognition. If the likelihood is Gaussian and p(x|z) is also Gaussian, one can directly marginalize x and work
with p(y|z) and p(z). The authors should at leas discuss the potential use of this method alongside with the BIGAN model which also provides a recognition model.
It is not clear how the HMC parameters are fixed.
The experiments do not have error bars (Figure 4.) This questions the significance of the results.
My overall impression is that there is little novelty in the proposed approach. Namely, using a GAN to learn the prior distribution, and then very well known
techniques to infer the original input image.
I have missed some references to related work on inverse problems. An example is:
https://arxiv.org/pdf/1712.03353.pdf
Is the original figure contained in the training set used to infer the GAN. If so that can lead to biased results.
I have missed a simple baseline in which one simply finds the training image that is closest to the corrupted observed or partially observed image.
My overall impression is that there is not much novelty in the paper as it is simply a combination of well known techniques. E.g. GANs and Bayesian inference with Monte Carlo methods.