Bayesian structured variable selection in linear regression models

<Para ID="Par1">In this paper we consider the Bayesian approach to the problem of variable selection in normal linear regression models with related predictors. We adopt a generalized singular <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$g$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>g</mi> </math> </EquationSource> </InlineEquation>-prior distribution for the unknown model parameters and the beta-prime prior for the scaling factor <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$g$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>g</mi> </math> </EquationSource> </InlineEquation>, which results in a closed-form expression of the marginal posterior distribution without integral representation. A special prior on the model space is then advocated to reflect and maintain the hierarchical or structural relationships among predictors. It is shown that under some nominal assumptions, the proposed approach is consistent in terms of model selection and prediction. Simulation studies show that our proposed approach has a good performance for structured variable selection in linear regression models. Finally, a real-data example is analyzed for illustrative purposes. Copyright Springer-Verlag Berlin Heidelberg 2015