Bayesian methods for subset selection and survival analysis

The selection of predictors to include is a crucial problem in building a multiple regression model. The canonical regression setup in multiple regression models assumes the normal errors, which is restricted. This thesis proposes and develops a procedure that relaxes this normal assumption to smooth unimodal and symmetric errors. This approach assumes the error density is a scale mixture of normal. The likelihood reduces to an explicit sum over partitions, and standard Bayesian arguments are applied to identify the best subset of variables. The posterior mode subset is used as an estimator of the best subset, and a weighted Chinese restaurant process (WCR) is implemented to compute posterior quantities. Further more, the variable selection method in multiple regressions can be extended to develop the Bayesian subset selection in the Cox proportional hazard model. In this thesis, the variable selection procedure is built on the mixture hazard rate model. Whereas variable selection has been a hot topic in terms of supervised learning, the unsupervised variable/feature selection has become the focus of much research as the large amount of available unlabeled data has exploded. In this thesis, we proposed a variable selection method via slight modification of the weighted Chinese restaurant process (WCR) and variable ranking criteria in a totally unsupervised fashion. Moreover, a Bayesian analysis of the proportional hazard model with right censored data is given. Different from most of the Bayesian methods which model cumulative hazard function or the distribution of survival time, we use gamma distribution to model the baseline hazard rate. We establish results for the estimated cumulative hazard function and survival function. A numerical example is presented with comparison to the results obtained by other methods