We address the problem of estimating the distribution of presumed i.i.d. observations within the framework of Bayesian statistics. We propose a new posterior distribution that shares some similarities with the classical Bayesian one. In particular, when the statistical model is exact, we show that this new posterior distribution concentrates its mass around the target distribution, just as the classical Bayes posterior would do. However, unlike the Bayes posterior, we prove that these concentration properties remain stable when the equidistribution assumption is violated or when the data are i.i.d. with a distribution that does not belong to our model but only lies close enough to it. The results we obtain are non-asymptotic and involve explicit numerical constants.[-]