Lower bounds for posterior rates with Gaussian process priors

TL;DR

This paper establishes lower bounds for the convergence rates of Gaussian process posterior distributions, complementing existing upper bounds, and determines the exact rates in various settings, including extensions to Riemann-Liouville priors.

Contribution

It provides the first lower bounds for posterior convergence rates with Gaussian process priors, matching upper bounds and extending results to a broader class of priors.

Findings

Lower bounds match existing upper bounds, confirming optimal rates.

Precise convergence rates are identified for Gaussian priors in multiple contexts.

Extension of upper-bound results to Riemann-Liouville priors with a continuous parameter family.

Abstract

Upper bounds for rates of convergence of posterior distributions associated to Gaussian process priors are obtained by van der Vaart and van Zanten in [14] and expressed in terms of a concentration function involving the Reproducing Kernel Hilbert Space of the Gaussian prior. Here lower-bound counterparts are obtained. As a corollary, we obtain the precise rate of convergence of posteriors for Gaussian priors in various settings. Additionally, we extend the upper-bound results of [14] about Riemann-Liouville priors to a continuous family of parameters.