Laplace Variational Inference for Dirichlet Process Mixtures of Marked Poisson Point Processes

Marked point process data arise when events occur in a space with event-level marks. We study clustering of replicated marked Poisson point processes and introduce Dirichlet process mixtures of marked Poisson point processes, a Bayesian nonparametric model that jointly infers latent cluster structure, the number of clusters, and continuous mark-specific intensity surfaces. We use a squared link intensity representation to obtain tractable continuous domain likelihood terms without gridding or thinning. For posterior inference, we develop an efficient variational Bayes algorithm with a constrained Laplace approximation for the nonconjugate basis-coefficient block. The resulting coefficient update is formulated as a constrained optimization problem, which avoids the sign ambiguity and nodal-line issue of squared-link models. We further establish theoretical guarantees for mode finding optimization. We demonstrate the performance of the proposed model and algorithm through synthetic experiments and real-data analysis.