Contraction properties of shrinkage priors in logistic regression

Abstract

Bayesian shrinkage priors have received a lot of attention recently because of their efficiency in computation and accuracy in estimation and variable selection. In this paper, we study the contraction properties of shrinkage priors in a logistic regression model where the number of covariates is high. For a shrinkage prior distribution that is heavy-tailed and concentrated around zero with high probability such as the horseshoe prior, the Dirichlet–Laplace prior, and the normal-gamma prior with appropriate choices of hyper-parameters, estimates of the logistic regression coefficient are shown to asymptotically concentrate around the true sparse vector in the L2-sense. It is shown that the proposed contraction rate is comparable with the point mass prior that is studied in Atchadé (2017). The simulation study under the logistic regression model verifies the theoretical results by showing that the horseshoe prior and the Dirichlet–Laplace prior perform like the point mass prior for the estimation, variable selection and prediction, and yield much better results than Bayesian lasso and the non-informative normal prior.