Posterior concentration rates for infinite dimensional exponential families

Frequentist properties of Bayesian nonparametric procedures have been increasingly studied in the last decade, following the seminal papers of Barron et al. [1] and Ghosal et al. [8] which established general conditions on the prior and on the true distribution to obtain posterior consistency for the former and posterior concentration rates for the latter. Consistency of the posterior distribution is admitted as a minimal requirement, both from a subjectivist and an objectivist view-point, see Diaconis and Freedman [6]. Studying posterior concentration rates allows for more refined results, in particular it helps in understanding some aspects of the prior and can be used to compare a Bayesian procedure with another Bayesian procedures together with frequentist procedures. In the frequentist nonparametric literature the