On Adaptive Bayesian Inference

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title = "On Adaptive Bayesian Inference",
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.",
keywords = "log spline density., density function, posterior distribution, rate of convergence, Adaptation",
author = "Yang Xing",
year = "2008",
doi = "10.1214/08-EJS244",
language = "English",
volume = "2",
pages = "848--863",
journal = "Electronic Journal of Statistics",
issn = "1935-7524",
publisher = "Institute of Mathematical Statistics",