Executive Summary : | Matrix-valued data is used in various scientific applications, including spatio-temporal analysis, multivariate time series analysis, radiological image analysis, and sports analytics. The matrix-variate Normal distribution is commonly used for inference but is insufficient for modeling data with heavy-tailed errors. This work proposes a Bayesian nonparametric mixture modeling approach using the matrix-t distribution to address this issue. The approach involves modeling observations in each cluster as a matrix-t, with a random number of clusters. The computational computations are performed using Markov chain Monte Carlo (MCMC) via a collapsed Gibbs sampler, exploiting the conjugate structure in the model for Bayesian inference. The theoretical aspects of the approach are explored, including the posterior contraction rate of the mixing measure and posterior consistency for the number of clusters. The method is proposed for analyzing data from functional magnetic resonance imaging (fMRI) scans of the brain, as well as multivariate spatio-temporal data. The authors also propose the development of Bayesian matrix-t graphical models (BMtGM), which can infer the brain network topology while accounting for temporal covariance structure in a high-dimensional framework. |