A Bayesian approach to selecting covariates for prediction (Q2722304)

An important problem when building a multiple regression model is how to decide which covariates, or predictors, should be included. The problem of selecting a regression model from a large class of possible models in the case where no true model is believed to exist is considered in the present paper. In most situations it is straightforward enough to assemble a set of predictors as candidates from which a subset must be chosen, so that if \(p\) such candidate predictors are available (including a constant term) the largest model to be entertained could be written as \(y=X\beta + \epsilon,\) where \(y\) is an \(n\times 1\) column vector of observations on a response variable, \(X\) is an \(n\times p\) matrix of observations on predictor variables and \(\epsilon\) is an \(n\times 1\) column vector of i.i.d. \(N(0, \sigma^{2})\) disturbances. If a constant term is to be included in the model, then in general a set of \(2^{p-1}\) candidate models, \(M_{i},\) \(i\in I\), say, are available and the problem faced by the investigator is which of these should be chosen.NEWLINENEWLINENEWLINEIn developing an approach to this problem the paper is concerned with the case in which the investigator does not believe that there is a ``true'' model but that nevertheless he wants to choose an approximate model from those that are available from which to predict. In this context a Bayesian predictive model selection technique is developed when proper conjugate priors are used and an easily computed expression for the model selection criterion is obtained. Expressions for updating the value of the statistic are derived when a predictor is dropped from the model and the approach is applied to a large well-known data set.