Bayesian Analysis for Over-parameterized Linear Model via Effective Spectra

In high-dimensional Bayesian statistics, various methods have been developed, including prior distributions that induce parameter sparsity to handle many parameters. Yet, these approaches often overlook the rich spectral structure of the covariate matrix, which can be crucial when true signals are not sparse. To address this gap, we introduce a data-adaptive Gaussian prior whose covariance is aligned with the leading eigenvectors of the sample covariance. This prior design targets the data's intrinsic complexity rather than its ambient dimension by concentrating the parameter search along principal data directions. We establish contraction rates of the corresponding posterior distribution, which reveal how the mass in the spectrum affects the prediction error bounds. Furthermore, we derive a truncated Gaussian approximation to the posterior (i.e., a Bernstein-von Mises-type result), which allows for uncertainty quantification with a reduced computational burden. Our findings demonstrate that Bayesian methods leveraging spectral information of the data are effective for estimation in non-sparse, high-dimensional settings.