Latent space models are popular for analyzing dynamic network data. We propose a variational approach to estimate the model parameters as well as the latent positions of the nodes in the network. The variational approach is much faster than Markov chain Monte Carlo algorithms, and is able to handle large networks. Theoretical properties of the variational Bayes risk of the proposed procedure are provided. We apply the variational method and latent space model to simulated data as well as real data to demonstrate its performance.
We develop a variational Bayesian (VB) approach for estimating large-scale dynamic network models in the network autoregression framework. The VB approach allows for the automatic identification of the dynamic structure of such a model and obtains a direct approximation of the posterior density. Compared to Markov Chain Monte Carlo (MCMC) based sampling approaches, the VB approach achieves enhanced computational efficiency without sacrificing estimation accuracy. In the simulation study conducted here, the proposed VB approach detects various types of proper active structures for dynamic network models. Compared to the alternative approach, the proposed method achieves similar or better accuracy, and its computational time is halved. In a real data analysis scenario of day-ahead natural gas flow prediction in the German gas transmission network with 51 nodes between October 2013 and September 2015, the VB approach delivers promising forecasting accuracy along with clearly detected structures in terms of dynamic dependence.
Continuous latent time series models are prevalent in Bayesian modeling; examples include the Kalman filter, dynamic collaborative filtering, or dynamic topic models. These models often benefit from structured, non mean field variational approximations that capture correlations between time steps. Black box variational inference with reparameterization gradients (BBVI) allows us to explore a rich new class of Bayesian non-conjugate latent time series models; however, a naive application of BBVI to such structured variational models would scale quadratically in the number of time steps. We describe a BBVI algorithm analogous to the forward-backward algorithm which instead scales linearly in time. It allows us to efficiently sample from the variational distribution and estimate the gradients of the ELBO. Finally, we show results on the recently proposed dynamic word embedding model, which was trained using our method.
A two-level group-specific curve model is such that the mean response of each member of a group is a separate smooth function of a predictor of interest. The three-level extension is such that one grouping variable is nested within another one, and higher level extensions are analogous. Streamlined variational inference for higher level group-specific curve models is a challenging problem. We confront it by systematically working through two-level and then three-level cases and making use of the higher level sparse matrix infrastructure laid down in Nolan and Wand (2018). A motivation is analysis of data from ultrasound technology for which three-level group-specific curve models are appropriate. Whilst extension to the number of levels exceeding three is not covered explicitly, the pattern established by our systematic approach sheds light on what is required for even higher level group-specific curve models.
The evolution of communities in dynamic (time-varying) network data is a prominent topic of interest. A popular approach to understanding these dynamic networks is to embed the dyadic relations into a latent metric space. While methods for clustering with this approach exist for dynamic networks, they all assume a static community structure. This paper presents a Bayesian nonparametric model for dynamic networks that can model networks with evolving community structures. Our model extends existing latent space approaches by explicitly modeling the additions, deletions, splits, and mergers of groups with a hierarchical Dirichlet process hidden Markov model. Our proposed approach, the hierarchical Dirichlet process latent position clustering model (HDP-LPCM), incorporates transitivity, models both individual and group level aspects of the data, and avoids the computationally expensive selection of the number of groups required by most popular methods. We provide a Markov chain Monte Carlo estimation algorithm and apply our method to synthetic and real-world networks to demonstrate its performance.
Gaussian state space models have been used for decades as generative models of sequential data. They admit an intuitive probabilistic interpretation, have a simple functional form, and enjoy widespread adoption. We introduce a unified algorithm to efficiently learn a broad class of linear and non-linear state space models, including variants where the emission and transition distributions are modeled by deep neural networks. Our learning algorithm simultaneously learns a compiled inference network and the generative model, leveraging a structured variational approximation parameterized by recurrent neural networks to mimic the posterior distribution. We apply the learning algorithm to both synthetic and real-world datasets, demonstrating its scalability and versatility. We find that using the structured approximation to the posterior results in models with significantly higher held-out likelihood.