Bayesian Shrinkage in Linear Models with an Application to Genetic Networks

Different challenging issues have emerged in recent years regarding the analysis of high dimensional data. In such datasets, the number of observations is much lower than the number of covariates which is problematic in the conventional statistical model. Nowadays, highdimensional dataset is common in several fields of sciences such as biology, economics, genetics and medicine. For instance, gene expression data is an example of the high-dimensional dataset where the number of genes is larger than the number of samples (patients). In order to tackle the issue of high-dimensional there are many regularization and shrinkage methods that have been proposed and developed to gain a sparse model; these approaches belong either to frequentist penalty methods or Bayesian shrinkage prior. In this thesis, we aim to overcome the high dimensional problem through proposing two alternative novel Bayesian shrinkage prior distributions. The suggested methods are hierarchical Normal inverse Pareto distribution (hNiP) and Rescaled Beta hierarchical Normal inverse Pareto distribution (ReB-hNiP). We consider that both proposed priors are absolutely continuous prior distributions and belong to the family of scale mixture of normal distribution. The proposed Bayesian shrinkage methods have been applied for three different linear models. Particularly, they have been applied and compared with some shrinkage methods based on multivariate Bayesian linear regression model. The proposed methods have also been implemented on both the Bayesian linear regression model with measurement error model a shrinkage and dynamic Bayesian networks with measurement error model. The hyperparameters were selected by using several criteria such as Watanabe- Akaike information criteria (WAIC).