## Convergence rates for posterior distributions and adaptive estimation.(English)Zbl 1095.62055

From the introduction: Bayesian methods have been used for nonparametric inference problems, and many theoretical results have been developed to investigate the asymptotic properties of nonparametric Bayesian methods. So far, the positive results are on consistency and convergence rates.
The goal of this paper is to develop theorems on convergence rates for posterior distributions which can be used for adaptive estimation. In this paper we have theorems on convergence rates in two contexts: density estimation and regression. In either case, we consider the Bayesian estimation of some function $$f$$ (a density function or a regression function) based on a sample $$(Z_1,\dots, Z_n)$$ and are interested in the convergence rates for the posterior distributions for $$f$$.

### MSC:

 62G20 Asymptotic properties of nonparametric inference 62G07 Density estimation 62G08 Nonparametric regression and quantile regression 62F15 Bayesian inference

### Keywords:

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