Paper:
Is it optimal to use the standard Gaussian prior in variational autoencoders? With Gaussian distributions, which are not weakly informative priors, variational autoencoders struggle to reconstruct the actual data. We provide numerical evidence that encourages using Student's t-distributions as default priors in variational autoencoders, and we challenge the usual setup for the variational autoencoder structure by comparing Gaussian and Student's t-distribution priors with different forms of the covariance matrix.

Review:
Summary:  This paper studies alternative priors for VAEs, comparing Normal distributions (diagonal and full covariance) against Student-t’s (diagonal and full covariance).   In particular, the paper is concerned with posterior collapse---i.e. posterior remains at the prior, limiting the model’s ability to reconstruct the data.  Experiments are performed on a synthetic 2D dataset, ‘Gaussian ovals,’ and on OMNIGLOT.  Results primarily take the form of visualizations of the reconstructed data and MSE / SSIM numbers. 
Pros:  Systematic study and investigation of alternative priors for deep generative models is an under-studied area.  Moreover, heavy-tailed priors such as the student-t---while widely successful for robust regression---have not been explored as extensively for latent variable models, to the best of my knowledge.  This paper makes some steps towards solving these open problems.  
Cons:  I have two primary critiques of the paper: (i) the experimental hypothesis is unclear, (ii) no engagement with the results of Mathieu et al. [ICML 2019], who also study Student-t priors for reconstruction (and disentanglement).  
Regarding (i), the paper seems to be testing two hypotheses simultaneously: the effect of diagonal vs full covariance matrices and exponential-tailed vs heavy-tailed priors.  The latter seems more crucial for purposes of reconstruction according to Figure 3 (since the diagonal St-t has good reconstruction).  Yet a proper study of the effect of tails vs reconstruction would report results as the degree of freedom parameter is gradually increased (as this controls the tails directly).  No careful ablation study of this sort is performed.  For comparison, see Figure 2 of Mathieu et al. [ICML 2019].  Moreover, the text simply calls the student-t a “weakly informative prior,” but it needs to be much more specific about what characteristics of the student-t are crucial.  If all we require is something “weakly informative,” why aren’t alternatives like a diffuse Gaussian or uniform also considered?
Regarding (ii), Mathieu et al. [ICML 2019] also study priors formed by products of student-t marginals, but their work is not cited.  Mathieu et al. [ICML 2019] also show that, due to student-t’s not being rotationally invariant (unlike the diagonal Gaussian), they improve disentanglement with only a minor degradation of reconstruction.  As this work also studies reconstruction in student-t VAEs, it should include some discussion of Mathieu et al. [ICML 2019]’s results---if not direct engagement with their hypotheses.       
Final Evaluation:  While I like the general motivation for this work, there are no clear experimental hypotheses being tested in the experiments.  
Mathieu, E., Rainforth, T., Narayanaswamy, S. and Teh, Y.W., 2018. Disentangling Disentanglement in Variational Autoencoders. ICML 2019.