Full adaptation to smoothness using randomly truncated series priors with Gaussian coefficients and inverse gamma scaling

TL;DR

This paper introduces a novel Bayesian series prior with random truncation, Gaussian coefficients, and inverse gamma scaling, achieving optimal adaptive posterior contraction rates for functional estimation.

Contribution

It develops a new prior framework that adapts to unknown smoothness levels and attains optimal convergence rates in nonparametric Bayesian inference.

Findings

Achieves posterior contraction at optimal rates

Demonstrates adaptation to arbitrary smoothness levels

Provides concrete examples in signal and drift estimation

Abstract

We study random series priors for estimating a functional parameter (f\in L^2[0,1]). We show that with a series prior with random truncation, Gaussian coefficients, and inverse gamma multiplicative scaling, it is possible to achieve posterior contraction at optimal rates and adaptation to arbitrary degrees of smoothness. We present general results that can be combined with existing rate of contraction results for various nonparametric estimation problems. We give concrete examples for signal estimation in white noise and drift estimation for a one-dimensional SDE.