Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$

Abstract

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking $L^r$-norm neighborhoods, $1\le r\le \infty$, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for $1\le r\le 2$, but deteriorate as $r$ increases beyond 2. In the case of Gaussian nonparametric regression a Gaussian prior is devised for which the posterior contracts at the optimal rate in all $L^r$-norms, $1\le r\le \infty$.

arXiv commentPublished in at http://dx.doi.org/10.1214/11-AOS924 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (8)

• Rates of contraction of posterior distributions based on Gaussian process priors
• Convergence rates of posterior distributions for noniid observations
• Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
• Reproducing kernel Hilbert spaces of Gaussian priors
• Posterior convergence rates of Dirichlet mixtures at smooth densities
• Global uniform risk bounds for wavelet deconvolution estimators
• On local $U$-statistic processes and the estimation of densities of functions of several sample variables
• Uniform limit theorems for wavelet density estimators

Cited by in corpus (17)

• Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems
• Posterior contraction in sparse Bayesian factor models for massive covariance matrices
• Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions
• On Bayesian supremum norm contraction rates
• Bayesian inverse problems with non-conjugate priors
• Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression
• On adaptive posterior concentration rates
• Bernstein -- von Mises theorems for statistical inverse problems II: Compound Poisson processes
• Nonparametric Bayesian inference for reversible multi-dimensional diffusions
• Borrowing strengh in hierarchical Bayes: Posterior concentration of the Dirichlet base measure
• Bayesian inverse problems with unknown operators
• Adaptive nonparametric drift estimation for diffusion processes using Faber-Schauder expansions
• Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient
• Nonparametric Bayesian estimation in a multidimensional diffusion model with high frequency data
• Spatiotemporal Besov Priors for Bayesian Inverse Problems
• Multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets
• Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets"