Objective Bayesian Variable Selection in 1 Well - Formulated Models 2 Short Title : Variable Selection in Well - Formulated Models

6 This paper investigates objective Bayesian variable selection when there is a hier7 archical dependence structure on the inclusion of predictors in the model, that, when 8 ignored, makes the variable selection dependent on how the predictors are coded. In 9 particular, we study the type of dependence found in polynomial response surfaces 10 of orders two and higher. We develop three classes of priors on the model space, 11 investigate both their operating characteristics and their effect on the variable selec12 tion outcome, and provide a Metropolis-Hastings algorithm for searching the space of 13 models. The tools proposed allow fast and thorough exploration of model spaces that 14 account for hierarchical polynomial structure in the predictors. Proposed choices for 15 the model priors allow for strong control over the number of false positives included in 16 “good” models. 17