Electronic Journal of Statistics

On adaptive Bayesian inference

Yang Xing

Full-text: Open access


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 nγ/(2γ+1) of convergence if the true density of the observations belongs to the Hölder space Cγ[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.

Article information

Electron. J. Statist., Volume 2 (2008), 848-862.

First available in Project Euclid: 23 September 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G20: Asymptotic properties
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

Adaptation rate of convergence posterior distribution density function log spline density


Xing, Yang. On adaptive Bayesian inference. Electron. J. Statist. 2 (2008), 848--862. doi:10.1214/08-EJS244. https://projecteuclid.org/euclid.ejs/1222175445

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