On consistency issues in Bayesian nonparametric testing - a review

Although there have been a lot of developpements in the recent years on estimation in Bayesian nonparametric models, from a theoretical point view as well as from a methodological point of view, little has been done on Bayesian testing in nonparametric frameworks. In this talk I will be interested on asymptotic properties of Bayesian tests when at least one of the hypotheses is nonparametric. I will first give some results on goodness of fit types of tests where one is interested in testing a parametric model against a nonparametric alternative embedding the parametric model. Then I will discuss the more delicate problem where both hypotheses are nonparametric. Such cases involve in particular tests for monotonicity, two-sample tests and estimation of the number of components in nonparametric mixture models. It will be shown that the Bayes factor or equivalently the 0-1 loss function might not be appropriate in such cases and that modifications need to be considered.