Statistics Ph.D. Dissertation Defense - Brandon Carter

This 2024 Dissertation Defense will be held on Friday, November 22 from 12:00 p.m. to 2:00 p.m. with Brandon Carter. This event will be hybrid. If you are able to attend in person, it will be held in WEL 5.204. If you need the Zoom link, please email stat.admin@austin.utexas.edu.

Title: Bayesian Spatial Models for Discrete Data: Methodological Advances with Applications in Urban Sociology

Advisor: Dr. Catherine Calder

Abstract: This dissertation addresses critical challenges in spatial modeling for discrete-valued outcomes over areal units, particularly within the Bayesian framework. First, we introduce a nonstationary spatial model for collective efficacy, an import sociological construct. Our approach incorporates administrative land-use data to improve fine-scale predictions of this sociological construct across Columbus, OH, using collective efficacy ratings collected from participants in the Adolescent Health and Development in Context (AHDC) Study. We propose a dimension-expanded latent spatial process and a land-use-based filter that efficiently connects the latent process to ordinal observed data, ensuring computationally efficient estimation and interpretable, spatially resolved predictions. Next, we explore Markov random fields (MRFs), with a specific focus on the Ising model and its phase transition properties. We develop prior predictive response functions—conceptually linked to thermodynamic quantities—as a systematic method for analyzing varying codings of MRFs and demonstrate how the centered autologistic model exhibits unique, non-standard behaviors under different spatial dependence configurations. Lastly, we propose a novel mixture of directed graphical models (MDGMs) as a computationally efficient and theoretically grounded alternative to traditional MRFs. This framework leverages directed acyclic graphs (DAGs) selected to match assumed spatial contiguity structures, preserving dependencies while offering flexible model fitting through three classes of compatible DAGs. Comprehensive simulation studies and real data applications in urban sociology underscore the strengths of the MDGM approach in comparison to existing MRF models, providing a new framework for Bayesian analysis of discrete-valued, spatially-referenced data.