We propose Bayesian methods for estimating the precision matrix in Gaussian graphical models. The methods lead to sparse and adaptively shrunk estimators of the precision matrix, and thus conduct model selection and estimation simultaneously. Our methods are based on selection and shrinkage priors leading to parsimonious parameterization of the precision (inverse covariance) matrix, which is essential in several applications in learning relationships among the variables. In Chapter I, we employ the Laplace prior on the off-diagonal element of the precision matrix, which is similar to the lasso model in a regression context. This type of prior encourages sparsity while providing shrinkage estimates. Secondly we introduce a novel type of selection prior that develops a sparse structure of the precision matrix by making most of the elements exactly zero, ensuring positive-definiteness. In Chapter II we extend the above methods to perform classification. Reverse-phase protein array (RPPA) analysis is a powerful, relatively new platform that allows for high-throughput, quantitative analysis of protein networks. One of the challenges that currently limits the potential of this technology is the lack of methods that allows for accurate data modeling and identification of related networks and samples. Such models may improve the accuracy of biological sample classification based on patterns of protein network activation, and provide insight into the distinct biological relationships underlying different cancers. We propose a Bayesian sparse graphical modeling approach motivated by RPPA data using selection priors on the conditional relationships in the presence of class information. We apply our methodology to an RPPA data set generated from panels of human breast cancer and ovarian cancer cell lines. We demonstrate that the model is able to distinguish the different cancer cell types more accurately than several existing models and to identify differential regulation of components of a critical signaling network (the PI3K-AKT pathway) between these cancers. This approach represents a powerful new tool that can be used to improve our understanding of protein networks in cancer. In Chapter III we extend these methods to mixtures of Gaussian graphical models for clustered data, with each mixture component being assumed Gaussian with an adaptive covariance structure. We model the data using Dirichlet processes and finite mixture models and discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalizing constants that are functions of parameters of interest which are a result of the restrictions on the correlation matrix. We evaluate the operating characteristics of our method via simulations, as well as discuss examples based on several real data sets.

The paper presents an Markov Chain Monte Carlo algorithm for both variable and covariance selection in the context of logistic mixed effects models. This algorithm allows us to sample solely from standard densities, with no additional tuning being needed. We apply a stochastic search variable approach to select explanatory variables as well as to determine the structure of the random effects covariance matrix. For logistic mixed effects models prior determination of explanatory variables and random effects is no longer prerequisite since the definite structure is chosen in a data-driven manner in the course of the modeling procedure. As an illustration two real-data examples from finance and tourism studies are given. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

We considered a non-centered parameterization of the standard random-effects model, which is based on the Cholesky decomposition of the variance-covariance matrix. The regression type structure of the non-centered parameterization allows to choose a simple, conditionally conjugate normal prior on the Cholesky factor. Based on the non-centered parameterization, we search for a parsimonious variance-covariance matrix by identifying the non-zero elements of the Cholesky factors using Bayesian variable selection methods. With this method we are able to learn from the data for each effect, whether it is random or not, and whether covariances among random effects are zero or not. An application in marketing shows a substantial reduction of the number of free elements of the variance-covariance matrix. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Networks have been widely used to describe interactions among objects in diverse fields. Given the interest in explaining a network by its structure, much attention has been drawn to finding clusters of nodes with dense connections within clusters but sparse connections between clusters. Such clusters are called communities, and identifying such clusters is known as community detection. Here, to perform community detection, I focus on stochastic blockmodels (SBM), a class of statistically-based generative models. I present a flexible SBM that represents different types of data as well as node attributes under a Bayesian framework. The proposed models explicitly capture community behavior by guaranteeing that connections are denser within communities than between communities. First, I present a degree-corrected SBM based on a logistic regression formulation to model binary networks. To fit the model, I obtain posterior samples via Gibbs sampling based on Polya-Gamma latent variables. I conduct inference based on a novel, canonically mapped centroid estimator that formally addresses label non-identifiability and captures representative community assignments. Next, to accommodate large-scale datasets, I further extend the degree-corrected SBM to a broader family of generalized linear models with group correction terms. To conduct exact inference efficiently, I develop an iteratively-reweighted least squares procedure that implicitly updates sufficient statistics on the network to obtain maximum a posteriori (MAP) estimators. I demonstrate the proposed model and estimation on simulated benchmark networks and various real-world datasets. Finally, I develop a Bayesian SBM for community-structured covariance selection. Here, I assume that the data at each node are Gaussian and a latent network where two nodes are not connected if their observations are conditionally independent given observations of other nodes. Under the context of biological and social applications, I expect that this latent network shows a block dependency structure that represents community behavior. Thus, to identify the latent network and detect communities, I propose a hierarchical prior in two levels: a spike-and-slab prior on off-diagonal entries of the concentration matrix for variable selection and a degree-corrected SBM to capture community behavior. I develop an efficient routine based on ridge regularization and MAP estimation to conduct inference.