Posterior model consistency in high-dimensional Bayesian variable selection with arbitrary priors

In the context of Bayesian regression modeling, posterior model consistency provides frequentist validation for Bayesian variable selection. A question that has long been open is whether posterior model consistency holds under arbitrary priors when high-dimensional variable selection is performed. In this paper, we aim to give an answer by establishing sufficient conditions for priors under which the posterior model distribution converges to a degenerate distribution at the true model. Our framework considers high-dimensional regression settings where the number of potential predictors grows at a rate faster than the sample size. We demonstrate that a wide selection of priors satisfy the conditions that we establish in this paper.