The varying coefficient models have been very important analytic tools to study the dynamic pattern in biomedicine fields. Since nonparametric varying coefficient models make few assumptions on the specification of the model, the 'curse of dimensionality' is an very important issue. Nonparametric Bayesian methods combat the curse of dimensionality through specifying a sparseness-favoring structure. This is accomplished through the Bayesian penalty for model complexity \citep{Jeffreys:Berger:1992} and is aided through centering on a base Bayesian parametric model. This dissertation presents three novel semiparametric Bayesian methods for the analysis of longitudinal data, diffusion tensor imaging data, and longitudinal circumplex data.

In longitudinal data analysis, we propose a semiparametric Bayes approach to allow the impact of the predictors to vary across subjects, which allows flexibly local borrowing of information across subjects. Local hypothesis testing and confidence bands are developed for the identification of time windows for significant predictor impact, adjusting for multiple comparisons. The methods are assessed using simulation studies and applied to a yeast cell-cycle gene expression data set.

In analyzing diffusion tensor imaging data, we propose a semiparametric Bayesian local functional model to connect multiple diffusion properties along white matter fiber bundles with a set of covariates of interest. An LPP2 prior facilitates global and local borrowing of information among subjects, while an infinite factor model flexibly represents low-dimensional structure. Local hypothesis testing and confidence bands are developed to identify fiber segments for significant association of covariates with multiple diffusion properties, controlling for multiple comparisons. The method is assessed by a simulation study and illustrated via two fiber tract data sets for neurodevelopment.

In analyzing longitudinal circumplex data, we propose a semiparametric Bayesian infinite state-space circumplex model to capture the dynamic transition pattern of affective experience, where affects are characterized as an ordering on the circumference of a circle. A sticky infinite state hidden Markov model via hierarchical Dirichlet proces is used to address the time related state-switching structure and the self-transition feature. The method is assessed by a simulation study and an emotion data set for the dynamics of emotion regulation.