The Annals of Statistics
- Ann. Statist.
- Volume 35, Number 2 (2007), 697-723.
Posterior convergence rates of Dirichlet mixtures at smooth densities
Subhashis Ghosal and Aad van der Vaart
Abstract
We study the rates of convergence of the posterior distribution for Bayesian density estimation with Dirichlet mixtures of normal distributions as the prior. The true density is assumed to be twice continuously differentiable. The bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order. In order to handle this problem, we derive a new general rate theorem by considering a countable covering of the parameter space whose prior probabilities satisfy a summability condition together with certain individual bounds on the Hellinger metric entropy. We apply this new general theorem on posterior convergence rates by computing bounds for Hellinger (bracketing) entropy numbers for the involved class of densities, the error in the approximation of a smooth density by normal mixtures and the concentration rate of the prior. The best obtainable rate of convergence of the posterior turns out to be equivalent to the well-known frequentist rate for integrated mean squared error n−2/5 up to a logarithmic factor.
Article information
Source
Ann. Statist. Volume 35, Number 2 (2007), 697-723.
Dates
First available in Project Euclid: 5 July 2007
Permanent link to this document
http://projecteuclid.org/euclid.aos/1183667289
Digital Object Identifier
doi:10.1214/009053606000001271
Mathematical Reviews number (MathSciNet)
MR2336864
Zentralblatt MATH identifier
1117.62046
Subjects
Primary: 62G07: Density estimation 62G20: Asymptotic properties
Keywords
Bracketing Dirichlet mixture entropy maximum likelihood mixture of normals posterior distribution rate of convergence sieve
Citation
Ghosal, Subhashis; van der Vaart, Aad. Posterior convergence rates of Dirichlet mixtures at smooth densities. Ann. Statist. 35 (2007), no. 2, 697--723. doi:10.1214/009053606000001271. http://projecteuclid.org/euclid.aos/1183667289.
