  We study the rate of convergence of posterior distributions in density
estimation problems for log-densities in periodic Sobolev classes characterized
by a smoothness parameter p. The posterior expected density provides a
nonparametric estimation procedure attaining the optimal minimax rate of
convergence under Hellinger loss if the posterior distribution achieves the
optimal rate over certain uniformity classes. A prior on the density class of
interest is induced by a prior on the coefficients of the trigonometric series
expansion of the log-density. We show that when p is known, the posterior
distribution of a Gaussian prior achieves the optimal rate provided the prior
variances die off sufficiently rapidly. For a mixture of normal distributions,
the mixing weights on the dimension of the exponential family are assumed to be
bounded below by an exponentially decreasing sequence. To avoid the use of
infinite bases, we develop priors that cut off the series at a
sample-size-dependent truncation point. When the degree of smoothness is
unknown, a finite mixture of normal priors indexed by the smoothness parameter,
which is also assigned a prior, produces the best rate. A rate-adaptive
estimator is derived.
