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We study the rate of convergence of posterior distributions in densityestimation problems for log-densities in periodic Sobolev classes characterizedby a smoothness parameter p. The posterior expected density provides anonparametric estimation procedure attaining the optimal minimax rate ofconvergence under Hellinger loss if the posterior distribution achieves theoptimal rate over certain uniformity classes. A prior on the density class ofinterest is induced by a prior on the coefficients of the trigonometric seriesexpansion of the log-density. We show that when p is known, the posteriordistribution of a Gaussian prior achieves the optimal rate provided the priorvariances die off sufficiently rapidly. For a mixture of normal distributions,the mixing weights on the dimension of the exponential family are assumed to bebounded below by an exponentially decreasing sequence. To avoid the use ofinfinite bases, we develop priors that cut off the series at asample-size-dependent truncation point. When the degree of smoothness isunknown, a finite mixture of normal priors indexed by the smoothness parameter,which is also assigned a prior, produces the best rate. A rate-adaptiveestimator is derived.

Convergence rates for Bayesian density estimation of infinite-dimensional exponential families