Propriety of the reference posterior distribution in Gaussian Process modeling

TL;DR

This paper extends the understanding of the propriety of the reference posterior in Gaussian Process models by proving its validity for smooth correlation kernels and analyzing tail behaviors.

Contribution

It provides a proof of the propriety of the reference posterior for smooth kernels like exponential with q=2 and Matérn with , expanding previous results.

Findings

Proves posterior propriety for smooth kernels such as exponential with q=2.

Establishes tail rates of the reference prior for these kernels.

Complements previous work on rough kernels by covering smooth cases.

Abstract

In a seminal article, Berger, De Oliveira and Sans\'o (2001) compare several objective prior distributions for the parameters of Gaussian Process regression models with isotropic correlation kernel. The reference prior distribution stands out among them insofar as it always leads to a proper posterior. They prove this result for rough correlation kernels - Spherical, Exponential with power $q<2$, Mat\'ern with smoothness $\nu<1$. This paper provides a proof for smooth correlation kernels - Exponential with power $q=2$, Mat\'ern with smoothness $\nu \geqslant 1$, Rational Quadratic - along with tail rates of the reference prior for these kernels.