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Abstract:
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We develop a general modeling strategy for performing locally adaptive Bayesian shrinkage of regression coefficients when the space of predictors is subject to heredity constraints. The predictors are assumed to form a directed acyclic graph, which provides a poset structure for covariate inclusion in the discrete model selection problem. We relax the strong and weak heredity conditions of the discrete model space into inequality relationships among the local shrinkage parameters of the regression coefficients of the full model. This strategy provides a means of incorporating local shrinkage in the form of a "horseshoe" prior while respecting the hierarchical structure of the predictors. Though the modeling strategy is motivated by the need to provide an efficient way to determine a good model for high order polynomial surface regression, we show that it can also be applied to non-parametric function and surface fitting when there is a natural structure to the basis functions used. We compare the methodology to discrete Bayesian model selection subject to heredity constraints as well as frequentist alternatives that are based on the LASSO.
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