Feller operators and mixture priors in Bayesian nonparametrics

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 to study 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 consistency of the corresponding posterior, extending results only known for specific kernels.