Statistica Sinica 26 (2016), 1175-1195

MULTISCALE BERNSTEIN POLYNOMIALS

FOR DENSITIES

Antonio Canale^{1,2} and David B. Dunson^{3}

Abstract: Our focus is on constructing a multiscale nonparametric prior for densities. The Bayes density estimation literature is dominated by single scale methods, with the exception of Polya trees, which favor overly-spiky densities even when the truth is smooth. We propose a multiscale Bernstein polynomial family of priors, which produce smooth realizations that do not rely on hard partitioning of the support. At each level in an infinitely-deep binary tree, we place a beta dictionary density; within a scale the densities are equivalent to Bernstein polynomials. Using a stick-breaking characterization, stochastically decreasing weights are allocated to the finer scale dictionary elements. A slice sampler is used for posterior computation, and properties are described. The method characterizes densities with locally-varying smoothness, and can produce a sequence of coarse to fine density estimates. An extension for Bayesian testing of group differences is introduced and applied to DNA methylation array data.

Key words and phrases: Density estimation, multiresolution, multiscale clustering, multiscale testing, nonparametric Bayes, Polya tree, stick-breaking, wavelets.