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April 2007 Posterior convergence rates of Dirichlet mixtures at smooth densities
Subhashis Ghosal, Aad van der Vaart
Ann. Statist. 35(2): 697-723 (April 2007). DOI: 10.1214/009053606000001271


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.


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Subhashis Ghosal. Aad van der Vaart. "Posterior convergence rates of Dirichlet mixtures at smooth densities." Ann. Statist. 35 (2) 697 - 723, April 2007.


Published: April 2007
First available in Project Euclid: 5 July 2007

zbMATH: 1117.62046
MathSciNet: MR2336864
Digital Object Identifier: 10.1214/009053606000001271

Primary: 62G07 , 62G20

Keywords: bracketing , Dirichlet mixture , Entropy , maximum likelihood , mixture of normals , posterior distribution , rate of convergence , sieve

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 2 • April 2007
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