Variational Generative Modeling of Stochastic Point Processes

We consider approximate inference for a class of Cox point processes i.e., point processes with stochastic intensities. Specifically, we consider processes where the Poisson intensity function is modeled as the solution of a stochastic differential equation (SDE). We propose a VAE-like approach where the latent variable is the solution to an SDE, the decoder is fixed (mapping the intensity function to a realization of an inhomogeneous Poisson process), and the encoder maps a point process realization to a posterior path measure corresponding to a diffusion process. Using tools from the theory of {\it enlargement of filtrations}, we show that the posterior path measure lies in a variational family of SDE path measures. Consequently, evidence lower bound (ELBO) maximization coincides with likelihood maximization. We also introduce hybrid encoder architectures for modeling the drift function of the posterior SDE, conditioned on varying length point process sample paths. Experiments on synthetic data showcase the ability to recover the ground truth measure and highlight the potential of this framework for modeling over-dispersed point processes