Adaptive density estimation based on a mixture of Gammas

TL;DR

This paper introduces an adaptive Bayesian density estimator on positive semiline using gamma mixture priors, achieving near-minimax convergence rates that adapt to the unknown smoothness of the density.

Contribution

It develops a hierarchical Bayesian approach with gamma mixture priors for density estimation, providing theoretical convergence rates and adaptivity to smoothness.

Findings

Achieves near-minimax posterior concentration rates

Constructs gamma mixture approximations for local Hölder densities

Demonstrates adaptivity to unknown smoothness levels

Abstract

We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local H\"older densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a log n term) and since the priors are independent of the smoothness the rates are adaptive to the smoothness.