Reproducing kernel Hilbert spaces of Gaussian priors

TL;DR

This paper reviews the properties of reproducing kernel Hilbert spaces linked to Gaussian processes, focusing on their role in nonparametric Bayesian inference and posterior contraction analysis.

Contribution

It provides a comprehensive overview of RKHS associated with Gaussian priors and introduces tools for analyzing posterior contraction rates in Bayesian nonparametrics.

Findings

Reproducing kernel Hilbert spaces are crucial for understanding Gaussian priors in Bayesian statistics.

Concentration functions in RKHS describe the rate of posterior contraction.

Series expansions and linear transformations of Gaussian variables aid in computing concentration functions.

Abstract

We review definitions and properties of reproducing kernel Hilbert spaces attached to Gaussian variables and processes, with a view to applications in nonparametric Bayesian statistics using Gaussian priors. The rate of contraction of posterior distributions based on Gaussian priors can be described through a concentration function that is expressed in the reproducing Hilbert space. Absolute continuity of Gaussian measures and concentration inequalities play an important role in understanding and deriving this result. Series expansions of Gaussian variables and transformations of their reproducing kernel Hilbert spaces under linear maps are useful tools to compute the concentration function.