Case Studies in Hierarchical Bayesian Modeling

My dissertation focuses on developing Bayesian methodology for complex data structures with an emphasis on building novel algorithms to reduce the computational complexity. One viewpoint of this dissertation is to develop a hierarchical model to detect change points from covariance valued time series data from the Human Connectome Project (HCP). The project provides an excellent source of neural data across different regions of interest (ROIs) of the living human brain. The standard approach to analyze the fMRI data is the generalized linear model (GLM) (Calhoun et al., 2001, 2004; Luo and Puthusserypady, 2008). Due to certain limitations such approaches (Glover, 2011; Turner, 2016), the dataset have been transformed into covariance matrices to represent individual specific functional connectivity (FC) over several time points. Individual specific data were available from an existing analysis (Dai et al., 2017) in the form of time varying covariance matrices representing the brain activity as the subjects perform a specific task. The FC represents the signal strengths of an individual while performing a task or in the resting state. These tasks are structured in a way for each person to switch to various activities at different time points. In chapter 2 and 3, I develop a methodology to find out whether the signal intensity changes during a task switch. As a preliminary objective of studying the heterogeneity of brain connectomics across the population, I develop a probabilistic model for a sample of covariance matrices using a scaled Wishart distribution. I stress here that our data units are available in the form of covariance matrices, and I use the Wishart distribution to create our likelihood function rather than its more common usage as a prior on covariance matrices. Based on empirical explorations suggesting the data matrices to have low effective rank, I further model the center of the Wishart distribution using an orthogonal factor model type decomposition. I encourage shrinkage towards a low rank structure through a novel shrinkage prior and discuss strategies to sample from the posterior distribution using a combination of Gibbs and slice sampling. I extend our modeling framework to a dynamic setting to detect change points. The efficacy of the approach is explored in various simulation settings and exemplified on several case studies including our motivating HCP data. Motivated by electronic health record data from Fels longitudinal study (FLS) (Roche, 1992), I come up with a monotone single index model to quantify the impact of certain health related measurements on percentage body fat In this chapter 4. The variable of interest percentage body fat (pbf) uniquely identifies the fat distribution and body composition which can be utilized in evaluations of inability, and mortality. PBF is also able to quantify the risk factor of cardiovascular and related disease (Forbes, 2012). I am interested to understand the pattern of the fat distribution over temporal domain and effects of certain covariates such as BMI, systolic and diastolic blood pressure etc. Our variable of interest is bounded in interval $(0,1)$ and Beta distributions are commonly used to model proportion valued response variables, commonly encountered in longitudinal studies. I develop semi-parametric Beta regression models for proportion valued responses using a logit link where the covariate effect is flexibly modeled using a interpretable monotone single index transform of a linear combination of covariates. Single index models are helpful for dimension reduction and can accommodate misspecification of the link function in generalized linear models. Our Bayesian methodology incorporates the response variables which are missing at random. I implement this methodology using Hamiltonian Monte Carlo (HMC) (Neal, 1994; Duane et al.,1987) aided by No-U-turn sampler (NUTS) (Hoffman and Gelman, 2014). I explore frequentist properties of our approach and assess the robustness with a detailed simulation study. Finally, I apply our model on the motivating dataset from human grow related study i.e. Fels longitudinal study and summaries our findings. In chapter 5, I develop a scalable algorithm in case of sampling from a structured Gaussian distribution. I provide a detailed discussion of the computational complexity of the algorithm and implementation of it in the high-dimensional cases under shrinkage priors. Such algorithms will be advantageous while developing a Bayesian methodology in case of genome wide association studies (GWAS)