Statistical process monitoring and hierarchical Bayesian modeling are two ways to learn more about processes of interest. In this work, we consider two main components: risk-adjusted monitoring and Bayesian hierarchical models for spatial data. Usually, if prior information about a process is known, it is important to incorporate this into the monitoring scheme. For example, when monitoring 30-day mortality rates after surgery, the pre-operative risk of patients based on health characteristics is often an indicator of how likely the surgery is to succeed. In these cases, risk-adjusted monitoring techniques are used. In this work, the practical limitations of the traditional implementation of risk-adjusted monitoring methods are discussed and an improved implementation is proposed. A method to perform spatial risk-adjustment based on exact locations of concurrent observations to account for spatial dependence is also described. Furthermore, the development of objective priors for fully Bayesian hierarchical models for areal data is explored for Gaussian responses. Collectively, these statistical methods serve as analytic tools to better monitor and model spatial processes. / Ph. D.

This dissertation consists of four independent but related parts, each in a Chapter. The first part is an introductory. It serves as the background introduction and offer preparations for later parts. The second part discusses two population multivariate normal distributions with common covariance matrix. The goal for this part is to derive objective/non-informative priors for the parameterizations and use these priors to build up constructive random posteriors of the Kullback-Liebler (KL) divergence of the two multivariate normal populations, which is proportional to the distance between the two means, weighted by the common precision matrix. We use the Cholesky decomposition for re-parameterization of the precision matrix. The KL divergence is a true distance measurement for divergence between the two multivariate normal populations with common covariance matrix. Frequentist properties of the Bayesian procedure using these objective priors are studied through analytical and numerical tools. The third part considers the star-shape Gaussian graphical model, which is a special case of undirected Gaussian graphical models. It is a multivariate normal distribution where the variables are grouped into one "global" group of variable set and several "local" groups of variable set. When conditioned on the global variable set, the local variable sets are independent of each other. We adopt the Cholesky decomposition for re-parametrization of precision matrix and derive Jeffreys' prior, reference prior, and invariant priors for new parameterizations. The frequentist properties of the Bayesian procedure using these objective priors are also studied. The last part concentrates on the discussion of objective Bayesian analysis for partial correlation coefficient and its application to multivariate Gaussian models. / Ph. D.