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 modeling a response variable. A selection criterion may be formulated by constructing an estimator of a measure known as a discrepancy function. Such a measure quantifies the disparity between the true model and a fitted candidate model, where can-didate models are defined 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 the fitted model. An optimal set of input variables is then determined by searching for the candidate model that minimizes the discrepancy function. Although this type of variable selection problem has been extensively studied, attention is less often paid to the problem of accounting for the uncertainty inherent to the model selection problem. 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. Key words: Cp statistic; discrepancy function; model selection criterion