  This paper investigates Bayesian variable selection when there is a
hierarchical dependence structure on the inclusion of predictors in the model.
In particular, we study the type of dependence found in polynomial response
surfaces of orders two and higher, whose model spaces are required to satisfy
weak or strong heredity conditions. These conditions restrict the inclusion of
higher-order terms depending upon the inclusion of lower-order parent terms. We
develop classes of priors on the model space, investigate their theoretical and
finite sample properties, and provide a Metropolis-Hastings algorithm for
searching the space of models. The tools proposed allow fast and thorough
exploration of model spaces that account for hierarchical polynomial structure
in the predictors and provide control of the inclusion of false positives in
high posterior probability models.
