Bayesian Computation with Application to Spatial Models and Neuroimaging

Abstract

Analysis of Neuroimaging data has experienced great strides over the last few decades. Two key aspects of Neuroimaging data are its high-dimensionality and complex spatio-temporal autocorrelation. Classical approaches are somewhat limited in dealing with these two issues, as a result, Bayesian approaches are being utilized more frequently due to their flexibility. Despite their flexibility, there are several challenges for Bayesian approaches with respect to the required computation. First, the need for an efficient posterior computation method is paramount. Second, even in conjugate models, statistical accuracy in Bayesian computation may be hard to achieve. Since accuracy is of primary concern when studying the human brain, a careful and innovative exploration of Bayesian models and computation is necessary.

In this dissertation, we address some of these issues by looking at various Bayesian computational algorithms in terms of both accuracy and speed in the context of Neuroimaging data. The algorithms we study are the Hamiltonian Monte Carlo (HMC), Variational Bayes (VB), and integrated nested Laplace approximation (INLA) algorithms. HMC is a MCMC method that's particularly powerful for sampling in high-dimensional space with highly correlated parameters. It's robust and accurate, yet not as fast as some approximate Bayesian methods, for example, Variational Bayes (VB). However, since there is no theoretical guarantee that the resulting posterior derived from VB is accurate, its performance has to be analyzed on a case-by-case basis. INLA is another extremely fast method based on numerical integration with Laplace approximations but, like VB, there are no generally applicable theoretical guarantees of accuracy.

In Chapter II we focus on a particular spatial point process model, namely the log Gaussian Cox Process (LGCP), and consider applications to ecological and neuroimaging data. Inference for the LGCP is challenging due to its non-conjugacy and doubly stochastic property. We develop HMC and VB algorithms for the LGCP model and make comparisons with INLA. In Chapter III, we turn our focus to the general linear model with autoregressive errors (GLM-AR) which is widely used in analyzing fMRI single subject data. We derive an HMC algorithm and compare it with the VB algorithm and the mass univariate approach using the Statistical Parametric Mapping (SPM) software program. In Chapter IV, we extend the original GLM-AR model to a new model where the order of the AR coefficients can varying spatially across the brain and call it GLM with spatially varying autoregressive orders (SVARO). Using simulations and real data we compare our SVARO model with GLM-AR model implemented under both our MCMC sampler and the SPM VB algorithm.

Our results shed light on several important issues. While HMC almost always yields the most accurate results, the performance of VB is strongly model specific. INLA is a fast alternative to MCMC methods but we observe some limitations when examining its accuracy in certain settings. Furthermore, our new SVARO model performs better than the GLM-AR model in a number of ways. Not surprisingly, more accurate algorithms generally require more computational time. By systematically evaluating the pros and cons of each method, we believe our work to be practically useful for those researchers considering the use of these methods.