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In the setting of nonparametric multivariate regression with unknown errorvariance, we study asymptotic properties of a Bayesian method for estimating aregression function f and its mixed partial derivatives. We use a random seriesof tensor product of B-splines with normal basis coefficients as a prior for f,and the error variance is either estimated using the empirical Bayes approachor is endowed with a suitable prior in a hierarchical Bayes approach. Weestablish pointwise, L2 and supremum norm posterior contraction rates for f andits mixed partial derivatives, and show that they coincide with the minimaxrates. Our results cover even the anisotropic situation, where the trueregression function may have different smoothness in different directions.Using the convergence bounds, we show that pointwise, L2 and supremum normcredible sets for f and its mixed partial derivatives have guaranteedfrequentist coverage with optimal size. New results on tensor products ofB-splines are also obtained in the course.

Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression