Title: Robust Bayes-Like Estimation: Rho-Bayes estimation,
Abstract: We consider the problem of estimating the joint distribution $P$ of $n$
independent random variables within the Bayes paradigm from a non-asymptotic
point of view. Assuming that $P$ admits some density $s$ with respect to a
given reference measure, we consider a density model $\overline S$ for $s$ that
we endow with a prior distribution $\pi$ (with support $\overline S$) and we
build a robust alternative to the classical Bayes posterior distribution which
possesses similar concentration properties around $s$ whenever it belongs to
the model $\overline S$. Furthermore, in density estimation, the Hellinger
distance between the classical and the robust posterior distributions tends to
0, as the number of observations tends to infinity, under suitable assumptions
on the model and the prior, provided that the model $\overline S$ contains the
true density $s$. However, unlike what happens with the classical Bayes
posterior distribution, we show that the concentration properties of this new
posterior distribution are still preserved in the case of a misspecification of
the model, that is when $s$ does not belong to $\overline S$ but is close
enough to it with respect to the Hellinger distance.