Adaptive Bayesian Density Estimation with Location-Scale Mixtures
Kruijer, Willem; Rousseau, Judith; Van Der Vaart, Aad (2009-05), Adaptive Bayesian Density Estimation with Location-Scale Mixtures, 7th Workshop on Bayesian Nonparametrics, 2009-06, Moncalieri, Italie
Type
Communication / ConférenceExternal document link
http://hal.archives-ouvertes.fr/hal-00389343/en/Date
2009-05Conference title
7th Workshop on Bayesian NonparametricsConference date
2009-06Conference city
MoncalieriConference country
ItalieMetadata
Show full item recordAbstract (EN)
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 $\beta$-H\"older, with squared integrable H\"older 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.Subjects / Keywords
Rate-adaptive density estimation; Bayesian density estimation; Nonparametric density estimation; Convergence Rates; Location-Scale MixturesRelated items
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Rousseau, Judith (2010) Article accepté pour publication ou publié
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