  In nonparametric regression problems involving multiple predictors, there is
typically interest in estimating an anisotropic multivariate regression surface
in the important predictors while discarding the unimportant ones. Our focus is
on defining a Bayesian procedure that leads to the minimax optimal rate of
posterior contraction (up to a log factor) adapting to the unknown dimension
and anisotropic smoothness of the true surface. We propose such an approach
based on a Gaussian process prior with dimension-specific scalings, which are
assigned carefully-chosen hyperpriors. We additionally show that using a
homogenous Gaussian process with a single bandwidth leads to a sub-optimal rate
in anisotropic cases.
