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Pré-Publication, Document De Travail Année : 2009

Adaptive Bayesian Density Estimation with Location-Scale Mixtures

Résumé

We study convergence rates of Bayesian density estimators based on finite location-scale mixtures of a kernel $C_p \exp\{-|x|^p\}$. We construct a finite mixture approximation of densities whose logarithm is locally $\beta$-Hölder, with squared integrable Hölder constant. Under additional tail and moment conditions, the approximation is minimax for both the supremum-norm and the Kullback-Leibler divergence. We use this approximation to establish convergence rates for a Bayesian mixture model with priors on the weights, locations, and the number of components. Regarding these priors, we provide general conditions under which the posterior converges at a near optimal rate, and is rate-adaptive with respect to the smoothness of $\log f_0$. Examples of priors which satisfy these conditions include Dirichlet and Polya-tree priors for the weights, and Poisson processes for the locations.
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Dates et versions

hal-00389343 , version 1 (28-05-2009)
hal-00389343 , version 2 (01-07-2010)

Identifiants

  • HAL Id : hal-00389343 , version 1

Citer

Willem Kruijer, Judith Rousseau, Aad A.W. van Der Vaart. Adaptive Bayesian Density Estimation with Location-Scale Mixtures. 2009. ⟨hal-00389343v1⟩
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