Asymptotic behaviour of credible regions
The reknown theorem of Bernstein von Mises in regular finite-dimensional models has numerous interesting consequences, in particular, it implies that a large class of credible regions are also asymptotically confidence regions, which in turns imply that different priors lead to the same credible regions to first order. Unfortunately, the Bernstein von Mises theorem does not necessarily hold in high or infinite-dimensional models and understanding the asymptotic behaviour of credible regions is much more involved. In this talk, I will describe what are the new advances that have been obtained over the last 6 years or so in this area and I will in particular discuss some interesting phenomena which have been exhibited in high dimensional models. In particular, I will discuss the behaviour of model choice types of priors, encountered for instance in mixture models with unknown number of components, in regression models with a large number of covariates etc… where we can show that in a significant number of cases these priors tend to over penalize (or over smooth), leading to non-robust confidence statements.