An Integration by Parts Formula for Functionals of the Dirichlet-Ferguson Measure, and Applications

The Dirichlet-Ferguson measure is a random probability measure that has seen widespread use in Bayesian nonparametrics. Our main results can be seen as a first step towards the development of a stochastic analysis of the Dirichlet-Ferguson measure. We define a gradient that acts on functionals of the measure and derive its adjoint. The corresponding integration by parts formula is used to prove a covariance representation formula for square integrable functionals of the Dirichlet-Ferguson measure and to provide a quantitative central limit theorem for the first chaos. Our findings are illustrated by a variety of examples.