This dissertation is devoted to modeling complex data from the

Bayesian perspective via constructing priors with latent structures.

There are three major contexts in which this is done -- strategies for

the analysis of dynamic longitudinal data, estimating

shape-constrained functions, and identifying subgroups. The

methodology is illustrated in three different

interdisciplinary contexts: (1) adaptive measurement testing in

education; (2) emulation of computer models for vehicle crashworthiness; and (3) subgroup analyses based on biomarkers.

Chapter 1 presents an overview of the utilized latent structured

priors and an overview of the remainder of the thesis. Chapter 2 is

motivated by the problem of analyzing dichotomous longitudinal data

observed at variable and irregular time points for adaptive

measurement testing in education. One of its main contributions lies

in developing a new class of Dynamic Item Response (DIR) models via

specifying a novel dynamic structure on the prior of the latent

trait. The Bayesian inference for DIR models is undertaken, which

permits borrowing strength from different individuals, allows the

retrospective analysis of an individual's changing ability...

Previous papers have elaborated formal gradient analysis for spatial processes, focusing on the distribution theory for directional derivatives associated with a response variable assumed to follow a Gaussian process model. In the current work, these ideas are extended to additionally accommodate one or more continuous covariate(s) whose directional derivatives are of interest and to relate the behavior of the directional derivatives of the response surface to those of the covariate surface(s). It is of interest to assess whether, in some sense, the gradients of the response follow those of the explanatory variable(s), thereby gaining insight into the local relationships between the variables. The joint Gaussian structure of the spatial random effects and associated directional derivatives allows for explicit distribution theory and, hence, kriging across the spatial region using multivariate normal theory. The gradient analysis is illustrated for bivariate and multivariate spatial models, non-Gaussian responses such as presence-absence and point patterns, and outlined for several additional spatial modeling frameworks that commonly arise in the literature. Working within a hierarchical modeling framework, posterior samples enable all gradient analyses to occur as post model fitting procedures.