Paper:
We introduce Neural Markov Logic Networks (NMLNs), a statistical relational learning system that borrows ideas from Markov logic. Like Markov Logic Networks (MLNs), NMLNs are an exponential-family model for modelling distributions over possible worlds, but unlike MLNs, they do not rely on explicitly specified first-order logic rules. Instead, NMLNs learn an implicit representation of such rules as a neural network that acts as a potential function on fragments of the relational structure. Interestingly, any MLN can be represented as an NMLN. Similarly to recently proposed Neural theorem provers (Rocktäschel & Riedel, 2017), NMLNs can exploit embeddings of constants but, unlike NTPs, NMLNs work well also in their absence. This is extremely important for predicting in settings other than the transductive one. We showcase the potential of NMLNs on knowledge-base completion tasks and on generation of molecular (graph) data.

Review:
The following paper provides an extension to Markov Logic Networks(MLNs), by removing their dependency on pre-defined first-order logic rules.  This is handled via neural networks which are able to capture the statistical relations, so-called Neural Markov Logic Networks(NMLNs). As this is an implicit representation from the neural network, the rules act as potential functions on the MLN structure. As general MLN techniques are reliant on domain experts or exhaustive structure learning approaches, NMLNs are able to model more domains as provided in the work with knowledge-base completion tasks and generative modelling of molecules. 
The primary contribution in this body of work is based on the observation that relational structure repeats regularities in the data, and where deriving the statistics of these regularities is what allows for improved accuracy in a model.  The proposed NMLN  is architecturally identical to MLNs with the difference being the addition of the potential function.
As defined by the paper, fragments are connected subsets derived from relational data. The authors derived sets of fragments with constants defined by values in the data and anonymized fragment sets with integer assignments. With potential functions sampled from the anonymized and the true value fragments. The objective is a search for a maximum-entropy distribution to model the data derived fragments. The neural network aspect comes in the form of the minimum-maximum entropy modelling with weights for given fragments being learned by minimising the entropy of the fragment potential function. Where the model also maximizes the log-likelihood related to the anonymized fragments. The intuition in this work is that by selecting the maximum entropy distribution while also minimizing it by selecting the most informative statistical information for it, we will derive an accurate probability distribution given the possible worlds.
Overall, the paper performs a well enough job explaining the technical aspects. It does a thorough job explaining the algorithmic detail in the main body, and the appendix provides clear and implementable pseudocode equations. It is not exactly clear to me why the anonymization of fragments is necessary, but the authors suggest this places a greater focus on the graph structure and minimizes the model acting differently with different constants. The min-max entropy modelling also appears to be a novel approach in terms of statistical relational modelling. The results also demonstrate the success of NMLNs modelling on relational data and KB completion.
Regarding the technical aspects, a few concerns are the claims that the domains for experimentation seem rather trivial since the smoking and nations dataset are common relational datasets, and the strength of this model is the ability to learn on other domains. This is possibly addressed with the molecule experiments, but more datasets would have helped in confirming the breadth of domains as claimed by the paper. It is also difficult to measure the success of their model with generative modelling as no baseline was present for the molecule experiments. I still view this paper as a positive contribution to MLN research, with a technique that is successful among the few experiments tested.
At the same time, however, given the emerging contributions in the area, such as relational neural networks (Poole et al.), although the technical development is sensible, I don’t find the contribution that technically novel, and seems somewhat straightforward. Given the preliminary investigations, this is a reject from my side.