Bayesian Nonparametric Approaches for the Analysis of Compositional Data Based on Bernstein Polynomials

• Activity Number: 619
• Type: Invited
• Date/Time: Thursday, August 13, 2015 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Alejandro Jara and Andrés F. Barrientos*
• Companies: Pontificia Universidad Católica de Chile and Duke University
• Keywords: Density estimation ; Density regression ; Random probability measures

Abstract

We discuss Bayesian nonparametric procedures for density estimation and fully nonparametric regression for compositional data, that is, data supported in a m-dimensional simplex. The procedures are based on modified classes of multivariate Bernstein polynomials. We show that the modified classes retain the well known approximation properties of the classical versions defined on an unit hypercube and on a simplex. Based on these classes, we define prior distributions on the space of all probability measures defined on a m-dimensional simplex. We show that the processes are well defined, have large support and the frequentist asymptotic behaviour of the posterior distribution is appropriated. Finally, novel classes of probability models for sets of predictor-dependent probability distributions are proposed. Appealing theoretical properties such as support, continuity, marginal distribution, correlation structure, and consistency of the posterior distribution are studied. A. Jara's work supported by FONDECYT 1141193 grant. A.F. Barrientos's work supported by NSF grant ACI 1443014.