Bayesian aspects of some nonparametric problems
Open Access
- 1 April 2000
- journal article
- Published by Institute of Mathematical Statistics in The Annals of Statistics
- Vol. 28 (2), 532-552
- https://doi.org/10.1214/aos/1016218229
Abstract
We study the Bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation. We consider the asymptotic properties of Bayes procedures for conjugate (= Gaussian) priors. We show that so long as the prior puts nonzero measure on the very large parameter set of interest then the Bayes estimators are not satisfactory. More specifically, we show that these estimators do not achieve the correct minimax rate over norm bounded sets in the parameter space. Thus all Bayes estimators for proper Gaussian priors have zero asymptotic efficiency in this minimax sense. We then present a class of priors whose Bayes procedures attain the optimal minimax rate of convergence. These priors may be viewed as compound, or hierarchical, mixtures of suitable Gaussian distributions.Keywords
This publication has 20 references indexed in Scilit:
- Wald Lecture: On the Bernstein-von Mises theorem with infinite-dimensional parametersThe Annals of Statistics, 1999
- Asymptotic equivalence of density estimation and Gaussian white noiseThe Annals of Statistics, 1996
- Asymptotic equivalence of nonparametric regression and white noiseThe Annals of Statistics, 1996
- An Analysis of Bayesian Inference for Nonparametric RegressionThe Annals of Statistics, 1993
- Renormalization Exponents and Optimal Pointwise Rates of ConvergenceThe Annals of Statistics, 1992
- Geometrizing Rates of Convergence, IIIThe Annals of Statistics, 1991
- Minimax Risk Over Hyperrectangles, and ImplicationsThe Annals of Statistics, 1990
- On the Consistency of Bayes EstimatesThe Annals of Statistics, 1986
- All Admissible Linear Estimators of the Mean of a Gaussian Distribution on a Hilbert SpaceThe Annals of Statistics, 1984
- Conjugate Priors for Exponential FamiliesThe Annals of Statistics, 1979