For the past five years, I have been working on problems related to two classes of statistical models: networks and point processes. These random structures are often used in situations where traditional statistical assumptions do not hold (e.g. the data are not independent or identically distributed), meaning extra care must be taken to develop and extend statistical theory and tools to these settings. This dissertation consists of four separate projects (Chapters 2--5) each tackling a different problem pertaining to the point processes, network models, or their interface.
Chapter 2 develops a new hierarchical Bayesian point process model for an application in forensic science. Chapter 3 develops a new point process-based framework for latent position network models---as well as supporting theory---to fill an important gap in the existing sparse network literature. Chapter 4 develops an efficient Monte Carlo algorithm for Bayesian inference of large latent position network models, and Chapter 5 develops an efficient Monte Carlo algorithm for Bayesian inference of high-dimensional point process models that are useful for analyzing networks of neural spike trains.
My defense talk will primarily concentrate on the computational developments contained in Chapters 4 and 5.
>Rob Kass (Co-advisor)
Cosma Shalizi (Co-advisor)
Jared Murray (University of Texas Austin)