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On Adaptive Bayesian Inference

Xing, Yang LU (2008) In Electronic Journal of Statistics 2. p.848-863
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... (More)
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. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
keywords
log spline density., density function, posterior distribution, rate of convergence, Adaptation
in
Electronic Journal of Statistics
volume
2
pages
848 - 863
publisher
Institute of Mathematical Statistics
ISSN
1935-7524
DOI
10.1214/08-EJS244
language
English
LU publication?
no
id
b3be2546-40ce-4f8f-9c1f-ead354b36240 (old id 1465047)
date added to LUP
2009-08-20 13:59:43
date last changed
2016-06-29 09:06:13
@article{b3be2546-40ce-4f8f-9c1f-ead354b36240,
  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.},
  author       = {Xing, Yang},
  issn         = {1935-7524},
  keyword      = {log spline density.,density function,posterior distribution,rate of convergence,Adaptation},
  language     = {eng},
  pages        = {848--863},
  publisher    = {Institute of Mathematical Statistics},
  series       = {Electronic Journal of Statistics},
  title        = {On Adaptive Bayesian Inference},
  url          = {http://dx.doi.org/10.1214/08-EJS244},
  volume       = {2},
  year         = {2008},
}