Coverage of Uniform Credible Regions for Regression Function and Derivatives

• Activity Number: 392
• Type: Invited
• Date/Time: Tuesday, August 11, 2015 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Willliam Weimin Yoo and Subhashis Ghoshal*
• Companies: North Carolina State University and North Carolina State University
• Keywords: Coverage ; credible region ; posterior contraction ; sup norm ; nonparametric regression ; tensor product B-splines

Abstract

For nonparametric multivariate regression, we propose a Bayesian method of estimating the regression function and its mixed partial derivatives. The function is expanded in tensor products of B-splines with and coefficients are given a normal prior. The variance is either estimated using empirical Bayes method or is endowed with an inverse- gamma prior. We establish pointwise, L2-and sup-norm posterior contraction rates for the regression function and its mixed partial derivatives, and show that they coincide with the minimax rates for anisotropic Holder class of functions. Using concentration inequalities for Gaussian processes, we show that for the function or its mixed derivatives, the posterior variation around the posterior mean mimics the variation of the posterior mean around the true target function, which allows us to construct confidence regions for these functions that cover the function uniformly in its argument and have optimal sizes. An application to estimation and uncertainty quantification for the mode of the regression function is considered. Simulation results support theoretical findings. Some new results on tensor product B-spines are obtained in course.