Multivariate Bayesian function estimation

Details Multivariate Bayesian function estimation Multivariate Bayesian function estimation

2006-03-06

arxiv.org/abs/math/0603136v1

Annals of Statistics 2005, Vol. 33, No. 6, 2967-2999

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053605000000705 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053605000000705

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.

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