  In this paper, we investigate the problem of computing Bayesian estimators
using Langevin Monte-Carlo type approximation. The novelty of this paper is to
consider together the statistical and numerical counterparts (in a general
log-concave setting). More precisely, we address the following question: given
$n$ observations in $\mathbb{R}^q$ distributed under an unknown probability
$\mathbb{P}_{\theta^\star}$ with $\theta^\star \in \mathbb{R}^d$ , what is the
optimal numerical strategy and its cost for the approximation of $\theta^\star$
with the Bayesian posterior mean? To answer this question, we establish some
quantitative statistical bounds related to the underlying Poincar\'e constant
of the model and establish new results about the numerical approximation of
Gibbs measures by Cesaro averages of Euler schemes of (over-damped) Langevin
diffusions. These last results include in particular some quantitative controls
in the weakly convex case based on new bounds on the solution of the related
Poisson equation of the diffusion.
