Bayesian density estimation for compositional data using random Bernstein polynomials

We propose a Bayesian nonparametric procedure for density estimation for data in a dd-dimensional simplex. To this aim, we propose a prior distribution on probability measures based on a modified class of multivariate Bernstein polynomials. The model for the probability distribution corresponds to a mixture of Dirichlet distributions, with random weights and a random number of components. Theoretical properties of the proposal are provided, including posterior consistency and concentration rates of the posterior distribution.