  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. Here we extend this
idea by considering projections onto subspaces defined via some form of $L_1$
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. We also consider use of the approach
in structured variable selection problems such as ANOVA models where it is
desired to incorporate main effects in the presence of interactions. Here we
make use of projections related to the non-negative garotte which are able to
respect the hierarchical constraints. We also prove a consistency result
concerning the posterior distribution on the model induced by the projection,
and show that for some projections related to the adaptive lasso and
non-negative garotte the posterior distribution concentrates on the true model
asymptotically.
