Bayes Variable Selection in Semiparametric Linear Models

There is a rich literature on Bayesian variable selection for parametric models. Our focus is on generalizing methods and asymptotic theory established for mixtures of <italic>g</italic>-priors to semiparametric linear regression models having unknown residual densities. Using a Dirichlet process location mixture for the residual density, we propose a semiparametric <italic>g</italic>-prior which incorporates an unknown matrix of cluster allocation indicators. For this class of priors, posterior computation can proceed via a straightforward stochastic search variable selection algorithm. In addition, Bayes' factor and variable selection consistency is shown to result under a class of proper priors on <italic>g</italic> even when the number of candidate predictors <italic>p</italic> is allowed to increase much faster than sample size <italic>n</italic>, while making sparsity assumptions on the true model size.