Paper:
Translation into morphologically-rich languages challenges neural machine translation (NMT) models with extremely sparse vocabularies where atomic treatment of surface forms is unrealistic. This problem is typically addressed by either pre-processing words into subword units or performing translation directly at the level of characters. The former is based on word segmentation algorithms optimized using corpus-level statistics with no regard to the translation task. The latter learns directly from translation data but requires rather deep architectures. In this paper, we propose to translate words by modeling word formation through a hierarchical latent variable model which mimics the process of morphological inflection. Our model generates words one character at a time by composing two latent representations: a continuous one, aimed at capturing the lexical semantics, and a set of (approximately) discrete features, aimed at capturing the morphosyntactic function, which are shared among different surface forms. Our model achieves better accuracy in translation into three morphologically-rich languages than conventional open-vocabulary NMT methods, while also demonstrating a better generalization capacity under low to mid-resource settings.

Review:
This paper addresses the problem of translating into morphologically-rich languages, which suffers from the problem of sparse vocabularies and high numbers of rare and unseen words. In particular, current approaches such as subwords lack explicit notions of morphology and are obtained independently of the translation objective, while operating at the character-level renders the learning of long-distance dependencies more difficult.
More concretely, this paper models the generation of target words in a stochastic, hierarchical process, where the morphological features are modelled as latent variables. At each target word, the model samples a vector representing its lemma, followed by a k-dimensional latent inflectional features (e.g. nominative or accusative). To induce sparsity in the inflectional features, the paper uses a "stretch-and-rectify" distribution (Louizos et al., 2018) using the Kumaraswamy distribution. The paper further applies a sparsity-inducing regulariser to encourage the inflectional features to take discrete values of "0" or "1". Parameter estimation is done by optimising a lower-bound on the marginal log-likelihood. Experiments on translations into three morphologically-rich languages, English-{Arabic, Czech, Turkish}, indicate fairly small but consistent and statistically significant gains in BLEU. Further analysis indicates improved perplexity upper bound on rare words compared to subword-based baselines, along with somewhat interpretable latent inflectional features.
Overall, this paper proposes an elegant solution to an important problem, and yields statistically significant BLEU improvements over the baselines. I have listed some pros and cons below, with a recommendation of "Weak Accept". I would be willing to raise my scores assuming my concerns are sufficiently addressed. 
Pros:
1. The proposed approach is easy-to-follow and explained clearly, and the paper is overall well-written.
2. Experiments are done on translation into three morphologically-rich languages with both concatenative and non-concatenative morphology, and results in small but mostly consistent BLEU improvements over the best subword, hierarchical, and character baselines for each language.
3. The proposed approach is general and can potentially be applied on top of other model architectures (e.g. Transformers) and other language generation problems, such as language modelling, summarisation, etc, although the experiments in the paper focus on NMT with a GRU-based architecture.
Cons:
1. It is still unclear how much computational overhead is introduced by the approach (in terms of both training and prediction times), and how scalable the approach is when applied to language pairs that have much bigger training sets. In Tables 4 and 5, it seems that the largest dataset is English-Turkish multi-domain (434K sentence pairs), which is fairly small compared to other datasets.
2. Regarding the feature variations result in section 4.4.4, it is hard to draw any conclusions about the latent inflectional features just based on varying the inflectional features at one position. I would suggest running the same experiment for multiple words at different positions, and see whether the same set of inflection features always results in similar inflections. This would better convince the reader that the latent inflectional features really are capturing useful morphological information, and that the target word generation process is appropriately controlled by the latent variable.
3. Since some of the experiments did not use the same standard splits (e.g. the multi-domain Turkish experiments), it would be nice to report how well external models (e.g. OpenNMT) would do in the authors' dataset split, to make sure that the reported numbers here are at least comparable to that. This would help ensure the credibility of the findings.
4. The effect of data size experiments (Section 4.4.2) for the character model seems somewhat counter-intuitive. Why would making the training set bigger (by incorporating data from a more diverse domain) make the character model worse? The explanation that "[the character model]'s capacity cannot cope with the increased amount of sparsity" does not seem satisfactory.  Why does adding more data result in increased sparsity? Furthermore, one can simply use more hidden units or deeper layers to mitigate this problem in the character-based model. Assuming the authors are controlling for model capacity, then the character model's hidden state can be made bigger (since the vocabulary size is lower for character models, thus the embedding layer and the softmax layer are by definition smaller than the subword-based model).
Minor suggestions:
1. Section 3.5 is not very clear; adding some figures or more explanation there would help understand how prediction is done.
2. It would be interesting to explore other potential metric for measuring morphological generalisation. For instance, at evaluation time, does the model predict words that it has never seen before (e.g. predicting "plays", even though the model has only seen "play" at training time) at a higher frequency than character/subword/hierarchical baselines? If yes, this would provide more evidence that the latent inflectional features are properly capturing morphological information.