Statistica Sinica

Sonia Petrone and Piero Veronese

Abstract:Priors for Bayesian nonparametric inference on a continuous curve are often defined through approximation techniques, e.g. basis-functions expansions with random coefficients. Using constructive approximations is particularly attractive, since it may facilitate the prior elicitation. With this motivation we study a class of operators, introduced by Feller, for the constructive approximation of a bounded real function. Feller operators have a simple, probabilistic structure. We prove that, when the random elements used in their construction are chosen in the natural exponential family, they have several properties of interest in statistical applications, and can be represented as mixtures of simple probability distribution functions. As a by-product, we give some new results on the natural exponential family. Our construction offers more insights on the role of mixtures in Bayesian nonparametrics. A fairly general class of mixture priors arises, which includes continuous, countable, or finite mixtures, with kernels suggested by the approximation scheme. This allows the study of theoretical properties in a unified setting; in particular, we give results on the Kullback-Leibler property for the proposed class of mixture priors, and on the consistency of the corresponding posterior, extending results known only for specific kernels.

Key words and phrases:Bernstein polynomials, fiducial densities, Kullback-Leibler support, mixture models, natural exponential family, weak consistency.