Article

Title: Posterior Contraction Rates of Density Derivative Estimation

Issue: Volume 79 Series A Part 2 Year 2017
Abstract
In this paper, we study the problem of Bayesian estimation of derivatives of a density function on the unit interval. We use a finite random series prior based on $B$-splines and study the asymptotic properties of the posterior distribution under the setting of fixed smoothness of the true function. We obtain the posterior contraction rate under both the $L_2$- and $L_{\infty}$-distances. The rate under $L_2$-distance agrees with the minimax optimal rate. This result is then extended to the estimation of a multivariate density function on the unit cube and its mixed partial derivatives using tensor product $B$-splines.