Paper:
Poisson-Gamma Dynamical Systems
We introduce a new dynamical system for sequentially observed multivariate count data. This model is based on the gamma-Poisson construction-a natural choice for count data-and relies on a novel Bayesian nonparametric prior that ties and shrinks the model parameters, thus avoiding overfitting. We present an efficient MCMC inference algorithm that advances recent work on augmentation schemes for inference in negative binomial models. Finally, we demonstrate the model's inductive bias using a variety of real-world data sets, showing that it exhibits superior predictive performance over other models and infers highly interpretable latent structure.

Review:
This paper considers modelling sequential count data, which arises in many real world applications but results in challenges in its analysis due to its high dimensionality, overdispersion and potentially non-linear dynamics governing its structure over time. The authors develop a novel Gamme-Poisson dynamical system (PGDS), assuming a latent sequence of Gamma variables and Poisson observations. They impose shrinkage priors that allow for automatic tuning of the latent state dimensionality and extend the model to a non-parametric approach, defining two novel stochastic processes. For inference in this model, the authors develop an MCMC algorithm with efficient sampling updates, making use of two equivalent generative processes of the model and auxiliary variables. The authors perform a series of experiments on different datasets, comparing the PGDS to the classic Gaussian linear dynamical system and the Gamma process dynamic Poisson Factor Analysis, showing that the  PGDS outperforms the other two models in a number of applications, in particular in cases with overdispersed data. The authors further illustrate the interpretability of the learned model parameters in regards to real world applications.
Technical quality:  This is a rigorous and well-written paper with a lot of technical detail. The details included in the main paper are sufficient to develop a general understanding of the methods used, with the supplementary material offering proofs and more details on the sampling scheme. The experiments show the model's applicability to a variety of real world datasets, which ties in nicely with the way the method was motivated.  Minor details: the introduction mentions an Extended Kalman filter as a non-Gaussian extension to the LDS. I think this generally refers to non-linear extensions of the LDS, rather than non-Gaussian ones. There is also a larger body of work in the neuroscience literature that has been concerned with extensions of the LDS to Poisson observations (PLDS), combining the expressive power of Gaussian latent dynamics with Poisson observations. This could be included in the literature review and might be a nice contrast to the approach presented in this paper.   Novelty/originality:  The paper makes notable novel contributions by introducing the PGLD together with novel stochastic processes.  Potential impact or usefulness:   Novel approaches to modelling sequential count data should be applicable to a variety of real world processes, such as social networks or Neuroscience.   Clarity and presentation:  The paper is well-written and coherent. At points there are small errors (e.g. line 76) or typos which would require corrections, but these are very minor.