Andrés Felipe Barrientos (Pontificia Universidad Católica de Chile)

Bayesian density estimation for compositional data using random Bernstein polynomials

We propose a Bayesian nonparametric model for single density estimation, for data in the p-dimensional simplex space, say S_p. The proposal is based on a particular class of multivariate Bernstein polynomials on S_p and extends the Dirichlet-Bernstein prior for density estimation, for data in a closed, bounded interval. The resulting model corresponds to expressing the density of the data as a particular mixture of Dirichlet distributions. We show that these mixtures can approximate uniformly any continuous density function on S_p. Considering several topologies, the measurability of the process used to induce the prior distribution is characterized. In addition, appealing theoretical properties such as, large support and consistency of the posterior distribution are established for the model. Finally, we give some directions about the use of this proposal to define probability models for a set of predictor-dependent probability distributions for data in S_p.