Concentration of Posterior Distributions with Misspecified Models

National audienceWe investigate the asymptotic properties of posterior distributions when the model is misspecified, i.e. it is comtemplated that the observations $x_1,\cdots,x_n$ might be drawn from a density in a family $\{h_\sigma,\sigma\in\Theta\}$ where $\Theta\subset\mathbb R^d$, while the actual distribution of the observations may not correspond to any of the densities $h_\sigma$. A concentration property around a fixed value of the parameter is obtained as well as concentration properties around the maximum likelihood estimate