Posterior consistency in linear models under shrinkage priors

TL;DR

This paper studies how the posterior distribution of regression coefficients in high-dimensional linear models behaves asymptotically, showing concentration around the true parameters under certain conditions and popular shrinkage priors.

Contribution

It establishes posterior concentration results in high-dimensional linear models with shrinkage priors, extending understanding of Bayesian inference in such settings.

Findings

Posterior distributions concentrate around true parameters under specified conditions.

Results hold for common shrinkage priors with sparsity assumptions.

Provides theoretical guarantees for high-dimensional Bayesian linear models.

Abstract

We investigate the asymptotic behavior of posterior distributions of regression coefficients in high-dimensional linear models as the number of dimensions grows with the number of observations. We show that the posterior distribution concentrates in neighborhoods of the true parameter under simple sufficient conditions. These conditions hold under popular shrinkage priors given some sparsity assumptions.