Multi-dimensional point patterns represent collections of events distributed across domains that can span space, time, or even replications. A significant challenge in modeling such data lies in capturing their structural dependencies and dynamic behavior. We propose a flexible and tractable Bayesian nonparametric framework for modeling multi-dimensional Poisson process intensities, with a particular focus on replicated and space-time point patterns. Our approach utilizes the Bernstein-Dirichlet process prior as a foundational building block for modeling intensity functions. In the context of replicated point patterns, we develop a prior model to flexibly estimate a shared baseline process hierarchically, capturing shared structural features across replicates by pooling their components. We demonstrate the inferential capabilities of our method on simulated data and also apply it to data from the Divvy bike-share system in Chicago. To accommodate temporal dependencies, we explore a dependent Dirichlet process prior to extend the model to dynamic settings. For space-time point patterns, we extend the Bernstein-Dirichlet process prior model to higher dimensions, enabling an efficient and tractable inference mechanism.

Event Host: Andrew Le, Ph.D. Student, Statistical Science 

Advisor: Athanasios Kottas