  Bayesian methods are developed for the multivariate nonparametric regression
problem where the domain is taken to be a compact Riemannian manifold. In terms
of the latter, the underlying geometry of the manifold induces certain
symmetries on the multivariate nonparametric regression function. The Bayesian
approach then allows one to incorporate hierarchical Bayesian methods directly
into the spectral structure, thus providing a symmetry-adaptive multivariate
Bayesian function estimator. One can also diffuse away some prior information
in which the limiting case is a smoothing spline on the manifold. This,
together with the result that the smoothing spline solution obtains the minimax
rate of convergence in the multivariate nonparametric regression problem,
provides good frequentist properties for the Bayes estimators. An application
to astronomy is included.
