This thesis describes my research work in past years in the Statistic Department of Iowa State University. There are several key statistical features common to the whole thesis. In the first place, all the statistical methods are developed taking a Bayesian perspective to conduct the statistical inference. A second common feature of the two main parts is that both correspond to high-dimensional problems. In the first case, because a large amount of information for a few individuals is available, and in the second part due to model space is really large which brings computational intractability issues. Finally, the response variable in all data used here is a positive count, in the first part, it is associated with the gene expression while in the second part it represents a number of automobile crashes.