Poisson-Gamma Dynamical Systems with Time-varying Transition Dynamics
Authors: Jiahao Wang, Yijun Wang, Nan Fang, Sikun Yang
Organizations: Department of Electrical and Computer Engineering, University of Arizona, US · School of Computing and Information Technology, Great Bay University, China · Tsinghua University, China · Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Systems
Abstract
Bayesian methodologies for handling count-valued time series have gained prominence due to their ability to infer interpretable latent structures and to estimate uncertainties. Among these Bayesian models, Poisson-Gamma Dynamical Systems (PGDSs) are proven to be effective in capturing the evolving dynamics underlying observed count sequences. However, the state-of-the-art PGDS still falls short in capturing the transition dynamics that are commonly observed in real-world count time series. To mitigate this limitation, a PGDS with time-varying transition kernel (TV-PGDS), is proposed to allow the underlying transition matrices to evolve over time. Three specifically-designed Dirichlet Markov chains (Dir-Dir, Dir-Gam-Dir, PR-Gam-Dir) are constructed to accommodate heterogeneous structural mutations within these dependencies. Leveraging Dirichlet-Multinomial-Beta data augmentation techniques, a fully-conjugate and efficient Gibbs sampler is developed to perform posterior simulation. Experiments show that, in comparison with related models, the proposed PGDS achieves improved predictive performance due to its capacity to learn time-varying dependency structure captured by the time-evolving transition matrices.
Dynamic knowledge graphs are ubiquitous in today's AI applications, as we represent molecular structures, social relationships, and language information using these graph models. As knowledge graphs evolve over time and are often noisy and incomplete, modeling their temporal and relational dependencies becomes crucial for downstream tasks. To address these challenges, this paper proposes PGRE (Poisson-Gamma Relational Evolution), a probabilistic model for modeling inter-relational dependencies in dynamic knowledge graphs. PGRE represents multi-relational temporal links via a Poisson-Bernoulli formulation. It introduces Gamma-distributed latent variables to capture entity-factor associations and cross-relation dependencies mediated by shared latent communities. A Gamma Markov process further models the temporal evolution of these latent variables, enabling principled characterization of relational dynamics. Experiments on benchmark datasets show that PGRE achieves competitive performance in link prediction, particularly in sparse settings, while revealing meaningful relational evolution patterns in dynamic knowledge graphs.
Effectively modeling non-stationary dynamics in probabilistic multivariate time series(MTS) forecasting requires balancing expressiveness with robustness. Existing parametric approaches benefit from strong inductive biases but lack flexibility, whereas deep generative models struggle to capture complex temporal dependencies without extensive data and computation. We introduce Parametric Prior Mapping (PPM), a framework that injects parametric structural priors into a generative modeling process. Specifically, PPM utilizes a parametric estimator to derive a dynamic, adaptive prior that guides the learning of a complex predictive distribution via a learnable mapping. This design allows the model to retain the efficiency of parametric methods while exploiting the expressive power of generative models. Trained with a hybrid objective, PPM yields precise forecasts with well-calibrated uncertainty estimates. Empirical results show that PPM outperforms existing baselines in handling non-stationary data, offering a superior trade-off between accuracy and computational efficiency. The code is available at https://github.com/ljl8336/PPM.
State-space models (SSMs) are the standard formalism for Bayesian treatment of dynamical systems, with natural applications in statistics, signal processing, and machine learning. Despite their importance in both theory and application, dynamical systems have proven difficult to incorporate in modern probabilistic programming languages (PPLs), making state-of-the-art methods less accessible to practitioners and introducing friction in following the "Bayesian workflow." We introduce dynestyx, a probabilistic programming library with first-class support for SSMs, including state-of-the-art methods in the estimation of both states and parameters. Through a single, unified interface, users may specify arbitrary priors for discrete-time or continuous-time dynamical systems, perform inference over mixed-effect data, and make state and parameter estimates with principled uncertainty quantification.