Bayesian Modeling Using Latent Structures (2012)

<p>This dissertation is devoted to modeling complex data from the</p><p>Bayesian perspective via constructing priors with latent structures.</p><p>There are three major contexts in which this is done -- strategies for</p><p>the analysis of dynamic longitudinal data, estimating</p><p>shape-constrained functions, and identifying subgroups. The</p><p>methodology is illustrated in three different</p><p>interdisciplinary contexts: (1) adaptive measurement testing in</p><p>education; (2) emulation of computer models for vehicle crashworthiness; and (3) subgroup analyses based on biomarkers.</p><p>Chapter 1 presents an overview of the utilized latent structured</p><p>priors and an overview of the remainder of the thesis. Chapter 2 is</p><p>motivated by the problem of analyzing dichotomous longitudinal data</p><p>observed at variable and irregular time points for adaptive</p><p>measurement testing in education. One of its main contributions lies</p><p>in developing a new class of Dynamic Item Response (DIR) models via</p><p>specifying a novel dynamic structure on the prior of the latent</p><p>trait. The Bayesian inference for DIR models is undertaken, which</p><p>permits borrowing strength from different individuals, allows the</p><p>retrospective analysis of an individual's changing ability, and</p><p>allows for online prediction of one's ability changes. Proof of</p><p>posterior propriety is presented, ensuring that the objective</p><p>Bayesian analysis is rigorous.</p><p>Chapter 3 deals with nonparametric function estimation under</p><p>shape constraints, such as monotonicity, convexity or concavity. A</p><p>motivating illustration is to generate an emulator to approximate a computer</p><p>model for vehicle crashworthiness. Although Gaussian processes are</p><p>very flexible and widely used in function estimation, they are not</p><p>naturally amenable to incorporation of such constraints. Gaussian</p><p>processes with the squared exponential correlation function have the</p><p>interesting property that their derivative processes are also</p><p>Gaussian processes and are jointly Gaussian processes with the</p><p>original Gaussian process. This allows one to impose shape constraints</p><p>through the derivative process. Two alternative ways of incorporating derivative</p><p>information into Gaussian processes priors are proposed, with one</p><p>focusing on scenarios (important in emulation of computer</p><p>models) in which the function may have flat regions.</p><p>Chapter 4 introduces a Bayesian method to control for multiplicity</p><p>in subgroup analyses through tree-based models that limit the</p><p>subgroups under consideration to those that are a priori plausible.</p><p>Once the prior modeling of the tree is accomplished, each tree will</p><p>yield a statistical model; Bayesian model selection analyses then</p><p>complete the statistical computation for any quantity of interest,</p><p>resulting in multiplicity-controlled inferences. This research is</p><p>motivated by a problem of biomarker and subgroup identification to</p><p>develop tailored therapeutics. Chapter 5 presents conclusions and</p><p>some directions for future research.</p>