Adaptive Bayesian Density Estimation with Location-Scale Mixtures

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.ou