Multiscale Bernstein polynomials for densities

TL;DR

This paper introduces a multiscale Bayesian density estimation method using Bernstein polynomials, which adaptively captures varying smoothness and avoids overly-spiky results common in existing approaches.

Contribution

It proposes a novel multiscale Bernstein polynomial prior that produces smooth density estimates without hard partitioning, extending to Bayesian group difference testing.

Findings

Produces smooth density estimates with locally-varying smoothness

Uses a stick-breaking construction for flexible weighting

Applied to DNA methylation data for group comparison

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.