High-dimensional posterior consistency for hierarchical non-local priors in regression

The choice of tuning parameter in Bayesian variable selection is a critical problem in modern statistics. Especially in the related work of nonlocal prior in regression setting, the scale parameter reflects the dispersion of the non-local prior density around zero, and implicitly determines the size of the regression coefficients that will be shrunk to zero. In this paper, we introduce a fully Bayesian approach with the pMOM nonlocal prior where we place an appropriate Inverse-Gamma prior on the tuning parameter to analyze a more robust model that is comparatively immune to misspecification of scale parameter. Under standard regularity assumptions, we extend the previous work where $p$ is bounded by the number of observations $n$ and establish strong model selection consistency when $p$ is allowed to increase at a polynomial rate with $n$. Through simulation studies, we demonstrate that our model selection procedure outperforms commonly used penalized likelihood methods in a range of simulation settings.