Bayesian projection approaches to variable selection in generalized linear models

A Bayesian approach to variable selection which is based on the expected Kullback-Leibler divergence between the full model and its projection onto a submodel has recently been suggested in the literature. For generalized linear models an extension of this idea is proposed by considering projections onto subspaces defined via some form of L₁ constraint on the parameter in the full model. This leads to Bayesian model selection approaches related to the lasso. In the posterior distribution of the projection there is positive probability that some components are exactly zero and the posterior distribution on the model space induced by the projection allows exploration of model uncertainty. Use of the approach in structured variable selection problems such as ANOVA models is also considered, where it is desired to incorporate main effects in the presence of interactions. Projections related to the non-negative garotte are able to respect the hierarchical constraints. A consistency result is given concerning the posterior distribution on the model induced by the projection, showing that for some projections related to the adaptive lasso and non-negative garotte the posterior distribution concentrates on the true model asymptotically.