Universal Models of Multivariate Temporal Point Processes.

With the rapidly increasing availability of event stream data there is growing interest in multivariate temporal point process models to capture both qualitative and quantitative features of this type of data. Recent research on multivariate point processes have focused in inference and estimation problems for restricted classes of models such as continuous time Bayesian networks, Markov jump processes, Gaussian Cox processes, and Hawkes Processes. In this paper, we study the expressive power and learnability of Graphical Event Models (GEMs) - the analogue of directed graphical models for multivariate temporal point processes. In particular, we describe a set of Graphical Event Models (GEMs) and show that this class can universally approximate any smooth multivariate temporal point process. We also describe a universal learning algorithm for this class of GEMs and show, under a mild set of assumptions, learnability results for both the dependency structures and distributions in this class. Our consistency results demonstrate the possibility of learning about both qualitative and quantitative dependencies from rich event stream data.