Bayesian Estimation of Prediction Error and Variable Selection in Linear Regression

An important statistical application is the problem of determining an appropriate set of input variables for modelling a response variable. In such an application, candidate models are characterized by which input variables are included in the mean structure. A reasonable approach to gauging the propriety of a candidate model is to define a discrepancy function through the prediction error associated with this model. An optimal set of input variables is then determined by searching for the candidate model that minimizes the prediction error. In this paper, we focus on a Bayesian approach to estimating a discrepancy function based on prediction error in linear regression. It is shown how this approach provides an informative method for quantifying model selection uncertainty.