Bayesian Analysis of Structured and Multidimensional Functional Data with Applications to Electroencephalography Experiments

Many modern biomedical studies record vast amounts of data on individual subjects. The observed data may often be conceptualized as arising from an underlying smooth stochastic process after discretization and contimation with noise. Data in this form may exhibit multidimensionality and complex structural features. For example, electroencephalography (EEG) records electrical activity in the brain over continuous time. Repeated trials of cognitive tasks in EEG experiments induce longitudinal \textit{and} and functional dimensions, complicating estimation and inference. Regularized estimation and rigorous uncertainty quantification is highly sought after in these settings. In this dissertation I leverage techniques from factor analysis, probabilistic principal components analysis, and Gaussian processes (GPs) in the Bayesian paradigm. These techniques are crucial to achieve simultaneous flexible estimation and adaptive regularization. Model performance and calibration is assessed through a series of numerical experiments. The proposed methods are applied to analyze a wide variety of biomedical data, including cognitive EEG experiments, global age-specific fertility rates, and sleep EEG