Adaptive Bayesian Density Estimation with Location-Scale Mixtures

Abstract : We study convergence rates of Bayesian density estimators based on finite location-scale mixtures of a kernel proportional to $\exp\{-|x|^p\}$. 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 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 the logarithm of the true density.
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Willem Kruijer, Judith Rousseau, Aad van der Vaart. Adaptive Bayesian Density Estimation with Location-Scale Mixtures. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2010, 4, pp.1225-1257. ⟨hal-00389343v2⟩

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