In Bayesian inference, the choice of prior has been of great interest. Subjective priors are ideal if sufficient information on priors is available. However, in practice, we cannot collect enough information on priors. Then objective priors are a good substitute for subjective priors. In this dissertation, an independent reference prior based on a class of objective priors is examined. It is a reference prior derived by assuming that the parameters are independent. The independent reference prior introduced by Sun and Berger (1998) is extended and generalized. We provide an iterative algorithm to derive the general independent reference prior. We also propose a sufficient condition under which a closed form of the independent reference prior is derived without going through the iterations in the iterative algorithm. The independent reference prior is then shown to be useful in respect of the invariance and the first order matching property. It is proven that the independent reference prior is invariant under a type of one-to-one transformation of the parameters. It is also seen that the independent reference prior is a first order probability matching prior under a sufficient condition. We derive the independent reference priors for various examples. It is observed that they are first order matching priors and the reference priors in most of the examples. We also study an independent reference prior in some types of non-regular cases considered by Ghosal (1997). / 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.