Bayesian Modeling Using Latent Structures

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, and

allows for online prediction of one's ability changes. Proof of

posterior propriety is presented, ensuring that the objective

Bayesian analysis is rigorous.

Chapter 3 deals with nonparametric function estimation under

shape constraints, such as monotonicity, convexity or concavity. A

motivating illustration is to generate an emulator to approximate a computer

model for vehicle crashworthiness. Although Gaussian processes are

very flexible and widely used in function estimation, they are not

naturally amenable to incorporation of such constraints. Gaussian

processes with the squared exponential correlation function have the

interesting property that their derivative processes are also

Gaussian processes and are jointly Gaussian processes with the

original Gaussian process. This allows one to impose shape constraints

through the derivative process. Two alternative ways of incorporating derivative

information into Gaussian processes priors are proposed, with one

focusing on scenarios (important in emulation of computer

models) in which the function may have flat regions.

Chapter 4 introduces a Bayesian method to control for multiplicity

in subgroup analyses through tree-based models that limit the

subgroups under consideration to those that are a priori plausible.

Once the prior modeling of the tree is accomplished, each tree will

yield a statistical model; Bayesian model selection analyses then

complete the statistical computation for any quantity of interest,

resulting in multiplicity-controlled inferences. This research is

motivated by a problem of biomarker and subgroup identification to

develop tailored therapeutics. Chapter 5 presents conclusions and

some directions for future research.