Concentration of posterior model probabilities and normalized L0 criteria

Abstract

We study frequentist properties of Bayesian and L0 model selection, with a focus on (potentially non-linear) high-dimensional regression. We propose a construction to study how posterior probabilities and normalized L0 criteria concentrate on the (Kullback-Leibler) optimal model and other subsets of the model space. When such concentration occurs, one also bounds the frequentist probabilities of selecting the correct model, type I and type II errors. These results hold generally, and help validate the use of posterior probabilities and L0 criteria to control frequentist error probabilities associated to model selection and hypothesis tests. Regarding regression, we help understand the effect of the sparsity imposed by the prior or the L0 penalty, and of problem characteristics such as the sample size, signal-to-noise, dimension and true sparsity. A particular finding is that one may use less sparse formulations than would be asymptotically optimal, but still attain consistency and often also significantly better finite-sample performance. We also prove new results related to misspecifying the mean or covariance structures, and give tighter rates for certain non-local priors than currently available.