A framework for posterior consistency in model selection

We develop a framework to help understand frequentist properties of Bayesian model selection, specifically its ability to select the (Kullback-Leibler) optimal model and portray model selection uncertainty. We outline its general basis and then focus on linear regression. The contribution is not proving consistency under given prior conditions but providing finite-sample rates that describe how model selection depends on the prior and problem characteristics such as sample size, signal-to-noise, problem dimension and true sparsity. A corollary proves a strong form of convergence for $L_0$ penalties and pseudo-posterior probabilities of interest for $L_0$ uncertainty quantification. These results unify and extend current Bayesian model selection literature and signal limitations, specifically that asymptotically optimal sparse priors can significantly reduce power even for moderately large $n$ and that less sparse priors can improve power trade-offs not adequately captured by asymptotic rates. These issues are compounded by the fact that model misspecification often causes an exponential drop in power, as we briefly study here. Our examples confirm these findings, underlining the importance of considering the data at hand's characteristics to judge the quality of model selection procedures, rather than relying purely on asymptotics.