  It is now known that an extended Gaussian process model equipped with
rescaling can adapt to different smoothness levels of a function valued
parameter in many nonparametric Bayesian analyses, offering a posterior
convergence rate that is optimal (up to logarithmic factors) for the smoothness
class the true function belongs to. This optimal rate also depends on the
dimension of the function's domain and one could potentially obtain a faster
rate of convergence by casting the analysis in a lower dimensional subspace
that does not amount to any loss of information about the true function. In
general such a subspace is not known a priori but can be explored by equipping
the model with variable selection or linear projection. We demonstrate that for
nonparametric regression, classification, density estimation and density
regression, a rescaled Gaussian process model equipped with variable selection
or linear projection offers a posterior convergence rate that is optimal (up to
logarithmic factors) for the lowest dimension in which the analysis could be
cast without any loss of information about the true function. Theoretical
exploration of such dimension reduction features appears novel for Bayesian
nonparametric models with or without Gaussian processes.
