Date of Graduation


Document Type


Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level



Mathematical Sciences


John Tipton

Committee Member

Qingyang Zhang

Second Committee Member

Avishek Chakraborty


Breast tumor, Gaussian mixture model, Polya-gamma data augmentation, Spatial autocorrelation, Spatial Gaussian mixture model, THz imaging


In the first chapter of this dissertation we give a brief introduction to Markov chain Monte Carlo methods (MCMC) and their application in Bayesian inference. In particular, we discuss the Metropolis-Hastings and conjugate Gibbs algorithms and explore the computational underpinnings of these methods. The second chapter discusses how to incorporate spatial autocorrelation in linear a regression model with an emphasis on the computational framework for estimating the spatial correlation patterns.

The third chapter starts with an overview of Gaussian mixture models (GMMs). However, because in the GMM framework the observations are assumed to be independent, GMMs are less effective when the mixture data exhibits spatial autocorrelation. To improve the performance of GMMs on spatially-correlated mixture data, chapter three describes a spatially correlated model that uses Gaussian process priors to account for the autocorrelation in the classifications. However, the inclusion of spatially correlated Gaussian processes results in a computational burden which is resolved by applying a P\`{o}lya-gamma data augmentation scheme that results in improved fit of the GMM in spatially correlated mixtures. Chapter three then compares the performance of the GMM and spatial GMM models on simulated data with and without spatial autocorrelation in the class labels. Both qualitative and quantitative model evaluation results support our assumption that the spatial GMM performs better when observation are spatially-autocorrelated.

Chapter four applies the spatial Gaussian mixture model from chapter three to data obtained from ongoing work that aims to improve the accuracy in breast cancer margin assessment using THz imaging technology. In particular, the Bayesian estimate of uncertainty in the posterior probability from the spatial GMM shows promise in addressing the primary clinical question of determining the cancerous tumor margins.