This dissertation mainly presents a novel Bayesian method for sparse functional data. Specifically, two models are proposed, one of which models all individual functions with a common smoothness and the other groups individual functions with heterogeneous smoothness. The proposed method utilizes the mixed effects model representation of the penalized splines for both the mean function and the individual functions. Given noninformative or weakly informative priors, Bayesian inference on the proposed models are developed and computations are done by using Markov Chain Monte Carlo (MCMC) methods. It has been shown that the proposed Bayesian methods perform well on irregularly spaced sparse functional data, where a traditional mixed eects model may often fail. This dissertation also includes a small section onorthogonal series functional estimation for density functions.Statistic
