In this paper we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive contraction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive contraction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse problems, both in the white noise sequence model and the nonparametric regression model, using priors based on the singular value decomposition of the operator, location-mixture priors and splines prior, and recover minimax adaptive contraction rates.

Abstract.In this paper, we review some recent results obtained in the context of Bayesian non and semiparametric models in terms of posterior concentration, Bernstein-von Mises theorems and tests. Then two specific cases are studied in more details. The first concerns tests for monotonicity and the second some asymptotic properties of empirical Bayes procedures.Résumé. Cet article est un article de revue et présente un certain nombre de résultats récents sur les propriété fréquentistes de procédures bayésiennes non et semiparamétriques. Nous donnons notamment des conditions permettant d'obtenir un théorème de Bernstein -von Mises pour des fonctionnelles de la densité, des résultats sur la consistance de la loi a posteriori lorsque la loi a priori dépend des données et enfin un test de monotonicité dans un modèle de régression nonparamétrique.