  Neyman-Scott processes (NSPs) are point process models that generate clusters
of points in time or space. They are natural models for a wide range of
phenomena, ranging from neural spike trains to document streams. The clustering
property is achieved via a doubly stochastic formulation: first, a set of
latent events is drawn from a Poisson process; then, each latent event
generates a set of observed data points according to another Poisson process.
This construction is similar to Bayesian nonparametric mixture models like the
Dirichlet process mixture model (DPMM) in that the number of latent events
(i.e. clusters) is a random variable, but the point process formulation makes
the NSP especially well suited to modeling spatiotemporal data. While many
specialized algorithms have been developed for DPMMs, comparatively fewer works
have focused on inference in NSPs. Here, we present novel connections between
NSPs and DPMMs, with the key link being a third class of Bayesian mixture
models called mixture of finite mixture models (MFMMs). Leveraging this
connection, we adapt the standard collapsed Gibbs sampling algorithm for DPMMs
to enable scalable Bayesian inference on NSP models. We demonstrate the
potential of Neyman-Scott processes on a variety of applications including
sequence detection in neural spike trains and event detection in document
streams.
