Supremum norm posterior contraction and credible sets for nonparametric multivariate regression

Abstract

In the setting of nonparametric multivariate regression with unknown error variance $\sigma^{2}$, we study asymptotic properties of a Bayesian method for estimating a regression function $f$ and its mixed partial derivatives. We use a random series of tensor product of B-splines with normal basis coefficients as a prior for $f$, and $\sigma$ is either estimated using the empirical Bayes approach or is endowed with a suitable prior in a hierarchical Bayes approach. We establish pointwise, $L_{2}$ and $L_{\infty}$-posterior contraction rates for $f$ and its mixed partial derivatives, and show that they coincide with the minimax rates. Our results cover even the anisotropic situation, where the true regression function may have different smoothness in different directions. Using the convergence bounds, we show that pointwise, $L_{2}$ and $L_{\infty}$-credible sets for $f$ and its mixed partial derivatives have guaranteed frequentist coverage with optimal size. New results on tensor products of B-splines are also obtained in the course.