Adaptive Bayesian Density Estimation with Location-Scale Mixtures

Willem Kruijer (1) , Judith Rousseau (1, 2, 3) , Aad A.W. van Der Vaart (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 Université Paris Dauphine-PSL
4 Department of Mathematics [Amsterdam]

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.

Mots clés: Rate-adaptive density estimation; Bayesian density estimation; Nonparametric density estimation; Convergence Rates; Location-Scale Mixtures

Domaines: Statistiques [math.ST]

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