Multiscale stick-breaking mixture models

Abstract: We introduce a family of multiscale stick-breaking mixture models for Bayesian nonparametric density estimation. The Bayesian nonparametric literature is dominated by single scale methods, exception made for P`olya trees and allied approaches. Our proposal is based on a mixture specification exploiting an infinitely-deep binary tree of random weights that grows according to a multiscale generalization of a large class of stick-breaking processes; this multiscale stick-breaking is paired with specific stochastic processes generating sequences of parameters that induce stochastically ordered kernel functions. Properties of this family of multiscale stick-breaking mixtures are described. Focusing on a Gaussian specification, a Markov Chain Montecarlo algorithm for posterior computation is introduced. The performance of the method is illustrated analyzing both synthetic and real data sets. The method is well-suited for data living in $\mathbb{R}$ and is able to detect densities with varying degree of smoothness and local features.