On universal Bayesian adaptation

Summary We consider estimating a probability density p based on a random sample from this density by a Bayesian approach. The prior is constructed in two steps, by first constructing priors on a collection of models each expressing a qualitative prior guess on the true density, and next combining these priors in an overall prior by attaching prior weights to the models. The purpose is to show that the posterior distribution contracts to the true distribution at a rate that is (nearly) equal to the rate that would have been obtained had only the model that is most suitable for the true density been used. We study special model weights that yield this adaptation property in some generality. Examples include minimal discrete priors and finite-dimensional models, with special attention to scales of Banach spaces, such as Hölder spaces, spline models, and classes of densities that are not uniformly bounded away from zero or infinity.