CONCENTRATION OF POSTERIOR DISTRIBUTIONS WITH MISSPECIFIED MODELS

We investigate the asymptotic properties of posterior distributions when the model is misspecified, i.e. it is comtemplated that the observations x1,..., xn might be drawn from a density in a family {hσ,σ ∈ Θ} where Θ ⊂ IRd, while the actual distribution of the observations may not correspond to any of the densities hσ. A concentration property around a fixed value of the parameter is obtained as well as concentration properties around the maximum likelihood estimate. 1. Introduction Let x1, x2, · · · be independent and identically distributed observa-tions on some topological space X, with common law Q on (X,B(X)), where B(Ω) denotes the borel σ-field of any topological space Ω. Throughout the paper, we assume that Q is absolutely continuous with respect to some probability ν on (X,B(X)) and we denote by q its density. Let {hσ,σ ∈ Θ} (the model) be a set of densities wit