Adaptive posterior rate of convergence for infinite-dimensional exponential families.

The problem of estimating probability densities on the unit interval whose log-functions belong to a periodic Sobolev space is studied adopting a Bayesian approach. A prior is constructed so that the posterior converges at optimal rate in the minimax sense under Hellinger loss whichever is the degree of smoothness of the log-density. Thus, the point-wise posterior expectation of the density function provides an optimal non-parametric adaptive estimation procedure.