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

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.