FELLER OPERATORS AND MIXTURE PRIORS IN BAYESIAN NONPARAMETRICS

Priors for Beyesian nonparametric inference oil 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 ehcitation. With this motivation we study it class of operators, introduced by Feller. for the constructive approximation of it bounded real function Feller operators have it simple; probabilistic structure We prove that, when the random elements used ill their construction are chosen ill the natural exponential family; they have several properties of interest ill statistical applications, and call be represented as mixtures of simple probability distribution functions As it by-product, we give some new results oil the natural exponential family, Our construction offers more insights oil the role of mixtures ill Bayesian nonparametrics A fairly general class of mixture priors arises, which Includes continuous: countable. or finite mixtures, with kernels suggested by the approximation scheme Thus allows the study of theoretical properties Ill a unified Setting; In particular. we give results oil the Kullback-Leibler property for the proposed class of mixture pilots. and oil the consistency of the corresponding posterior, extending results known only for specific kernels