Paper:
Inferring causal effects of a treatment, intervention or policy from observational data is central to many applications. However, state-of-the-art methods for causal inference seldom consider the possibility that covariates have missing values, which is ubiquitous in many real-world analyses. Missing data greatly complicate causal inference procedures as they require an adapted unconfoundedness hypothesis which can be difficult to justify in practice. We circumvent this issue by considering latent confounders whose distribution is learned through variational autoencoders adapted to missing values. They can be used either as a pre-processing step prior to causal inference but we also suggest to embed them in a multiple imputation strategy to take into account the variability due to missing values. Numerical experiments demonstrate the effectiveness of the proposed methodology especially for non-linear models compared to competitors.

Review:
Summary:
       The paper considers average treatment effect estimation treatment T and an unobserved confounder Z causes the outcome Y with an added constraint that the observed X is a noisy measurement of the underlying Z and some of the entries of observed X are missing at random (MAR). Previous work (Kallus et al. 2018) on similar settings assumed a low rank model connecting Z and X along with some entries missing at random which we do not observe. Further, Y (outcome) is related to the treatment and Z with a linear model. They actually show that matrix factorization techniques with these assumptions form an unbiased estimator for the Average Treatment Effect. There is prior work on doubly robust estimators under ignorability assumptions.
In this paper, the authors want to consider general non-linear relationships between Z and X with the same MAR (missing at random assumption) for missing entries.  So they fit a latent variable model in the form of VAE to find the P(Z| observed part of X) using a slightly modified version of the ELBO Lower Bound. For missing entries, they just replace those entries by a constant and do the usual VAE fit. After the VAE fit, multiple Z's are samples from the optimized P(Z| observed X) and then used in the doubly robust formula on each Z for estimating the average treatment effect and then finally the estimates averaged over the different Z's.
There is an alternative where the conditional mean of the latent variable is estimated from the VAE and used in the doubly robust computation.
In many synthetic examples, authors compare this with existing baselines and show that their method improves.
Pros: 
 - Baselines compares are comprehensive enough from my perspective. 
 - The paper is well written with clear pointer to existing work on doubly robust estimators with standard ignorability assumptions and the work for the linear, low rank model case by Kallus et al. 2018.
Cons:
   Major Issues
     - There is no reason to believe that even for the synthetic experiments, that the VAE posteriors would asymptotically yield unbiased ATE's which is provably the case in (Kallus et al. 2018) (of course for their restricted linear model/low rank assumptions). There is no reason to suppose Z's used in eq (9) from the VAE satisfy ignorability even in the asymptotic limit. In this light, the paper in essence just estimates Z's from some latent model that is fit and then use those latents to regress Y and then computes ATE. This seems a natural heuristic to try given such a problem. So I don't find a big methodological novelty. I would be willing to increase my scores if the authors could convince me on this point.
  - For the LRMF model (Fig 4) the MF approach seems to do as well as the authors proposal and we have a guarantee for the MF case under those linear/low rank modeling assumptions. So the only demonstrated benefit is for the synthetic experiments for Fig 5 and Fig 3 (I agree that it is considerable particularly with large fraction of missing values in Fig 3) in whose settings we dont know about how unbiased it is in the limit. More synthetic experiments with different kind of generative models could be more convincing.
- Some real world data set would have been more convincing - although I agree ground truth is hard to come by.
Minor Issues:
   - You have set B=200 for all the experiments for MDC-MI. Do the results change when B is increased or decreased ?? Does variance go down or the bias itself changes with B -  This would be a useful insight to have.
  -  There is typo in the ELBO lower bound equation in page 11. There are other minor typos. Please correct for it.
 - Since the paper is about estimate treatment effect from measurements of an unobserved confounder - it is important to cite - https://ftp.cs.ucla.edu/pub/stat_ser/r366-reprint.pdf from the causal DAG literature. 
 - The covariance of X given Z for the DLVM model is not clear - It seems to say exp ( or some matrix vector products) * Identity. What does this mean ? 
- The feature dimensions seems to be set at 10 - so would we expect the same results in much higher dimensions - like say 100s for few tens of thousands of samples??
********UPDATE after reading the rebutall,changes and the new experiment**********
I appreciate the authors actually accepting that identifiability issues cannot be easily resolved even if one knows P*(Z|X).
I recommend the authors to elaborate on this point in the camera ready version. However, showing that the proposed methods work on a real benchmark (semi-synthetic one used in Shalit et. al 2017) is commendable.  However, I find that the MF method is competitive (almost all the time) with their method when Doubly robust estimators are used.
But having matched an existing baseline (the MF method) that deals with confounders on real data and showing superior synthetic results and authors clarifying and toning down their theoretical claims, I am inclined to increase the score to Weak accept.