The Annals of Statistics

Convergence rates for Bayesian density estimation of infinite-dimensional exponential families

Catia Scricciolo

Full-text: Open access

Abstract

We study the rate of convergence of posterior distributions in density estimation problems for log-densities in periodic Sobolev classes characterized by a smoothness parameter p. The posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under Hellinger loss if the posterior distribution achieves the optimal rate over certain uniformity classes. A prior on the density class of interest is induced by a prior on the coefficients of the trigonometric series expansion of the log-density. We show that when p is known, the posterior distribution of a Gaussian prior achieves the optimal rate provided the prior variances die off sufficiently rapidly. For a mixture of normal distributions, the mixing weights on the dimension of the exponential family are assumed to be bounded below by an exponentially decreasing sequence. To avoid the use of infinite bases, we develop priors that cut off the series at a sample-size-dependent truncation point. When the degree of smoothness is unknown, a finite mixture of normal priors indexed by the smoothness parameter, which is also assigned a prior, produces the best rate. A rate-adaptive estimator is derived.

Article information

Source
Ann. Statist. Volume 34, Number 6 (2006), 2897-2920.

Dates
First available in Project Euclid: 23 May 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1179935069

Digital Object Identifier
doi:10.1214/009053606000000911

Mathematical Reviews number (MathSciNet)
MR2329472

Zentralblatt MATH identifier
1114.62043

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62G07: Density estimation

Keywords
Bayesian adaptive density estimation infinite-dimensional exponential family posterior distribution rate of convergence sieve prior

Citation

Scricciolo, Catia. Convergence rates for Bayesian density estimation of infinite-dimensional exponential families. The Annals of Statistics 34 (2006), no. 6, 2897--2920. doi:10.1214/009053606000000911. http://projecteuclid.org/euclid.aos/1179935069.


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