  We study the rates of convergence of the posterior distribution for Bayesian
density estimation with Dirichlet mixtures of normal distributions as the
prior. The true density is assumed to be twice continuously differentiable. The
bandwidth is given a sequence of priors which is obtained by scaling a single
prior by an appropriate order. In order to handle this problem, we derive a new
general rate theorem by considering a countable covering of the parameter space
whose prior probabilities satisfy a summability condition together with certain
individual bounds on the Hellinger metric entropy. We apply this new general
theorem on posterior convergence rates by computing bounds for Hellinger
(bracketing) entropy numbers for the involved class of densities, the error in
the approximation of a smooth density by normal mixtures and the concentration
rate of the prior. The best obtainable rate of convergence of the posterior
turns out to be equivalent to the well-known frequentist rate for integrated
mean squared error $n^{-2/5}$ up to a logarithmic factor.
