On adaptive Bayesian inference

Details On adaptive Bayesian inference On adaptive Bayesian inference

2008-05-23

arxiv.org/abs/0805.3584v2

Electronic Journal of Statistics 2008, Vol. 2, 848-862

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/08-EJS244 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by t

Scientific paper

10.1214/08-EJS244

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.

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