Frequentist coverage and adaptation of nonparametric Bayesian credible sets

In Bayesian nonparametrics, posterior distributions for functional parameters are typically visualized by plotting the center’’ of the posterior distribution, for instance the posterior mean or mode, together with upper and lower bounds indicating an $\alpha$-credible set, i.e. a set that contains a large fraction $\alpha$ of the posterior mass (typically $\alpha = 0.95$). The credible bounds are intended to visualize the remaining uncertainty in the estimate. From a frequentist perspective however, it is in general not clear what the quality of such uncertainty quantifications is. It is well known that in infinite-dimensional models, Bayesian credible sets are not automatically frequentist confidence sets, in the sense that under the assumption that the data are in actual fact generated by a true parameter’’, it is not automatically true that they contain that truth with probability at least $\alpha$.

In a number of recent papers it was shown that Bayesian credible sets typically have good frequentist coverage when we are undersmoothing, i.e. we are using a prior that is less regular than the truth, and the coverage can be very bad if we are oversmoothing. These results are of limited use however since they are non-adaptive, in the sense that the methods for which good coverage was shown require knowledge of the regularity of the truth. In this talk we discuss new results that state that (slightly enlarged) credible sets corresponding to certain rate-adaptive Bayes procedures are automatically honest, adaptive confidence sets for large sets of thruths.

This is joint work with Botond Szabo and Aad van der Vaart.

This talk is part of the Statistics series.