Bayesian fMRI Data Analysis and Bayesian Optimal Design

The present dissertation consists of the work done on two projects. The first project deals with a Bayesian hierarchical multi-subject multiscale analysis of functional magnetic resonance imaging (fMRI) data. The second project deals with Bayesian optimal sequential design for random function estimation with the cubic spline prior.;As part of the first project, we develop methodology for Bayesian hierarchical multi-subject multiscale analysis of functional Magnetic Resonance Imaging (fMRI) data. We begin by modeling the brain images temporally with a standard general linear model (GLM). After that, we transform the resulting estimated standardized regression coefficient maps through a discrete wavelet transformation (DWT) to obtain a sparse representation in the wavelet space. Subsequently, we assign to the wavelet coefficients a prior that is a mixture of a point mass at zero and a Gaussian white noise. In this mixture prior for the wavelet coefficients, the mixture probabilities are related to the pattern of brain activity across different resolutions. To incorporate this information, we assume that the mixture probabilities for wavelet coefficients at same location and level are common across subjects. Furthermore, we assign for the mixture probabilities a prior that depends on few hyperparameters. We develop empirical Bayes methodology to estimate the hyperparameters and, as these hyperparameters are shared by all subjects, we obtain precise estimated values. Then we carry out inference in the wavelet space and obtain smoothed images of the regression coefficients by applying the inverse wavelet transform to the posterior mean of the wavelet coefficients. An application to computer simulated synthetic data has shown that, when compared to single subject analysis, our multi-subject methodology performs better in terms of mean squared error (MSE) and ROC curve based analysis. Finally, we illustrate the utility and flexibility of our multi-subject methodology with an application to an event-related fMRI dataset generated by Postle (2005) through a multi-subject fMRI study of working memory related brain activation.;As part of the second project, we develop a novel computational framework for Bayesian optimal sequential design for random function estimation. This computational framework is based on evolutionary Markov chain Monte Carlo, which combines ideas of genetic or evolutionary algorithms with the power of Markov chain Monte Carlo. Our framework is able to consider general models for the observations, such as exponential family distributions and scale mixtures of normals. In addition, our framework allows optimality criteria with general utility functions that may include competing objectives, such as for example minimization of costs, minimization of the distance between true and estimated functions, and minimization of the prediction error. Finally, we illustrate our novel methodology with an application to experimental design for a nonparametric regression problem with the cubic spline prior distribution.