This paper investigates Bayesian variable selection when there is ahierarchical dependence structure on the inclusion of predictors in the model.In particular, we study the type of dependence found in polynomial responsesurfaces of orders two and higher, whose model spaces are required to satisfyweak or strong heredity conditions. These conditions restrict the inclusion ofhigher-order terms depending upon the inclusion of lower-order parent terms. Wedevelop classes of priors on the model space, investigate their theoretical andfinite sample properties, and provide a Metropolis-Hastings algorithm forsearching the space of models. The tools proposed allow fast and thoroughexploration of model spaces that account for hierarchical polynomial structurein the predictors and provide control of the inclusion of false positives inhigh posterior probability models.
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