# On Adaptive Bayesian Inference

Forskningsoutput: TidskriftsbidragArtikel i vetenskaplig tidskrift

### Standard

I: Electronic Journal of Statistics, Vol. 2, 2008, s. 848-863.

Forskningsoutput: TidskriftsbidragArtikel i vetenskaplig tidskrift

### RIS

TY - JOUR

T1 - On Adaptive Bayesian Inference

AU - Xing, Yang

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - log spline density.

KW - density function

KW - posterior distribution

KW - rate of convergence

U2 - 10.1214/08-EJS244

DO - 10.1214/08-EJS244

M3 - Article

VL - 2

SP - 848

EP - 863

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

SN - 1935-7524

ER -