Multiscale stick-breaking mixture models

Bayesian nonparametric density estimation is dominated by single-scale methods, typically exploiting mixture model specifications, exception made for Pólya trees prior and allied approaches. In this paper we focus on developing a novel family of multiscale stick-breaking mixture models that inherits some of the advantages of both single-scale nonparametric mixtures and Pólya trees. 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 Monte Carlo algorithm for posterior computation is introduced. The performance of the method is illustrated analyzing both synthetic and real datasets consistently showing competitive results both in scenarios favoring single-scale and multiscale methods. The results suggest that the method is well suited to estimate densities with varying degree of smoothness and local features.