Bayesian optimal adaptive estimation using a sieve prior

TL;DR

This paper establishes general conditions for the convergence rates of Bayesian posterior distributions in nonparametric models using sieve priors, with applications to various statistical models and an analysis of loss function impacts.

Contribution

It introduces a novel combination of standard conditions to derive posterior contraction rates and addresses rate adaptation in complex parameter spaces like Sobolev classes.

Findings

Posterior contraction rates are derived for multiple models.

Adaptive Bayesian methods can be suboptimal under certain loss functions.

Lower bounds on rates are established for pointwise loss.

Abstract

We derive rates of contraction of posterior distributions on nonparametric models resulting from sieve priors. The aim of the paper is to provide general conditions to get posterior rates when the parameter space has a general structure, and rate adaptation when the parameter space is, e.g., a Sobolev class. The conditions employed, although standard in the literature, are combined in a different way. The results are applied to density, regression, nonlinear autoregression and Gaussian white noise models. In the latter we have also considered a loss function which is different from the usual l2 norm, namely the pointwise loss. In this case it is possible to prove that the adaptive Bayesian approach for the l2 loss is strongly suboptimal and we provide a lower bound on the rate.