Posterior contraction in sparse generalized linear models. (14th September 2020)

• Record Type: Journal Article
• Authors: Jeong, Seonghyun; Ghosal, Subhashis
• Is Part Of: Biometrika. Volume 108:Number 2(2021)
• Page Start: 367
• Page End: 379
• Publication Date: 2020-09-14
• Subjects: Fractional posterior -- Generalized linear model -- High-dimensional regression -- Posterior contraction rate -- Sparsity-inducing prior; Biometry -- Periodicals; 570.1519505
• DOI: 10.1093/biomet/asaa074
• Languages: English
• ISSNs: 0006-3444

Abstract

Summary: We study posterior contraction rates in sparse high-dimensional generalized linear models using priors incorporating sparsity. A mixture of a point mass at zero and a continuous distribution is used as the prior distribution on regression coefficients. In addition to the usual posterior, the fractional posterior, which is obtained by applying Bayes theorem with a fractional power of the likelihood, is also considered. The latter allows uniformity in posterior contraction over a larger subset of the parameter space. In our set-up, the link function of the generalized linear model need not be canonical. We show that Bayesian methods achieve convergence properties analogous to lasso-type procedures. Our results can be used to derive posterior contraction rates in many generalized linear models including logistic, Poisson regression and others.