About the non-asymptotic behaviour of Bayes estimators

This paper investigates the nonasymptotic properties of Bayes procedures for estimating an unknown distribution from $n$ i.i.d. observations. We assume that the prior is supported by a model $(\scr{S},h)$ (where $h$ is the Hellinger distance) with suitable metric properties involving the number of small balls that are needed to cover larger ones. We also require that the prior put enough probability on small balls but we do not assume that the true distribution belongs to the model which is therefore only viewed as an approximation of the truth. We only need to have a control of the Kullback-Leibler Information between the true distribution and an approximating one in the model in order to derive our risk bounds.