On Rates of Convergence for Bayesian Density Estimation

Abstract.  We consider the problem of estimating a compactly supported density taking a Bayesian nonparametric approach. We define a Dirichlet mixture prior that, while selecting piecewise constant densities, has full support on the Hellinger metric space of all commonly dominated probability measures on a known bounded interval. We derive pointwise rates of convergence for the posterior expected density by studying the speed at which the posterior mass accumulates on shrinking Hellinger neighbourhoods of the sampling density. If the data are sampled from a strictly positive, α -Hölderian density, with α  ∈ ( 0,1] , then the optimal convergence rate n^{− α / (2 α +1)} is obtained up to a logarithmic factor. Smoothing histograms by polygons, a continuous piecewise linear estimator is obtained that for twice continuously differentiable, strictly positive densities satisfying boundary conditions attains a rate comparable up to a logarithmic factor to the convergence rate n ^−4/5 for integrated mean squared error of kernel type density estimators.