Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density

TL;DR

This paper analyzes the convergence rates of Bayesian mixture models using Betas for density estimation on [0,1], demonstrating minimax optimality and adaptivity for smooth densities.

Contribution

It introduces a Beta mixture model with a novel parametrization and proves its posterior concentrates at the minimax rate, achieving adaptive density estimation.

Findings

Posterior concentrates at the minimax rate for Hölder densities.

The model is adaptive to unknown smoothness levels.

Results are relevant for both Bayesian and frequentist frameworks.

Abstract

In this paper, we investigate the asymptotic properties of nonparametric Bayesian mixtures of Betas for estimating a smooth density on $[0,1]$. We consider a parametrization of Beta distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while putting a prior on this scaling parameter. We prove that such Bayesian nonparametric models have good frequentist asymptotic properties. We determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a H\"{o}lder class with regularity $\beta$, for all positive $\beta$, leading to a minimax adaptive estimating procedure of the density. We also believe that the approximating results obtained on these mixtures of Beta densities can be of interest in a frequentist framework.