On adaptive Bayesian inference

TL;DR

This paper investigates the convergence rates of Bayesian posterior distributions with hierarchical priors, extending existing results to almost sure convergence and demonstrating optimal rates for log spline densities in nonparametric density estimation.

Contribution

It extends previous Bayesian consistency results to almost sure convergence and establishes optimal convergence rates for log spline density models under certain smoothness conditions.

Findings

Achieves the minimax rate $n^{-rac{eta}{2eta+1}}$ for density estimation in Hölder spaces.

Provides conditions for posterior concentration around the best model index.

Strengthens existing results on Bayesian convergence rates.

Abstract

We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general in-probability theorems on the rate of convergence of the resulting posterior distributions. We extend their results to almost sure assertions. As an application we study log spline densities with a finite number of models and obtain that the Bayes procedure achieves the optimal minimax rate $n^{-\gamma/(2\gamma+1)}$ of convergence if the true density of the observations belongs to the H\"{o}lder space $C^{\gamma}[0,1]$. This strengthens a result in [1; 2]. We also study consistency of posterior distributions of the model index and give conditions ensuring that the posterior distributions concentrate their masses near the index of the best model.