Model selection is among the most fundamental and commonly encountered statistical challenges in scientific research. Standard assumptions incorporated into Bayesian model selection procedures result in model selection procedures that are not competitive with commonly used penalized likelihood methods. We propose modifications of standard Bayesian methods by imposing non-local prior densities on model parameters. We show that the resulting model selection procedures are consistent in linear model settings when the number of possible covariates p is bounded by the number of observations n, a property that has not been extended to other model selection procedures. In addition to consistently identifying the true model, the proposed procedures provide accurate estimates of the posterior probability that each identified model is correct. Through simulation studies, we demonstrate that these model selection procedures perform as well or better than commonly used penalized likelihood methods in a range of simulation settings. Proofs of the primary theorem and corollaries underlying the sampling properties of the proposed procedures are provided in supplemental material that is available online.
Johnson, Valen and Rossell, David, "Bayesian Model Selection in High-dimensional Settings" (May 2011). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 67.