Bayesian Hierarchical Models with Latent Conjugate Multivariate Random Effects

Recent advances in computing and measurement technologies have led to an explosion in the amount of data that are being collected in many areas of application. Much of these data have network or graph structures, and they are common in diverse scientific areas, such as environment, biology, computer science, and sociology, among other areas. This dissertation makes two contributions towards inference problems in Bayesian statistical network analysis. We are specifically interested in the setting where the edges (that connect nodes in the network) are referenced over covariates and spatial locations. Chapter 1 of the dissertation focus on areal data analysis, and a literature review is presented in Section 1.1. Traditional conditional autoregressive (CAR) models use neighborhood information to define the adjacency matrix. Specifically, the neighborhoods are formed deterministically using the boundaries between the regions. However, areas far apart may be highly correlated, and this use of CAR models does not account for this structure. We propose a class of prior distributions for adjacency matrices, which can detect a relationship between two areas that do not share a boundary. Our approach is fully Bayesian, and involves a computationally efficient conjugate update of the adjacency matrix. To illustrate the high performance of our Bayesian hierarchical model, we present a simulation study, and an example using data made publically available by the New York City Department of Health. In Chapter 2, We develop a Bayesian approach to analyzing high-dimensional spatial point patterns. In particular, we extend the Log-Gaussian Cox Process (LGCP) model, where we replace Gaussian distributed random effects with multivariate log-gamma (MLG) random effects. Fitting a traditional LGCP model can be computational intensive and often requires the use of Metropolis-Hastings when implementing a Markov Chain Monte Carlo (MCMC). Our model uses the MLG distribution which leads to conjugate full-conditional distributions within a Gibbs sampler that are computational straightforward to simulate from. Additionally, the LGCP often requires a subjective specification of a discretization of the spatial domain. We propose the use of prior distribution on this choice, which can be efficiently incorporated into our model through the spatial change of support. We demonstrate the proposed methodology through simulated examples and motivating analyses of Starbucks locations. In the final chapter of this dissertation we give a brief conclusion of this thesis.