Propriety of the reference posterior distribution in Gaussian process modeling

Abstract

In a seminal article, Berger, De Oliveira and Sansó [J. Amer. Statist. Assoc. 96 (2001) 1361–1374] compare several objective prior distributions for the parameters of Gaussian process 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 $\mathit{\rho }<2$, Matérn with smoothness $\mathit{\nu }<1$. This paper provides a proof for smooth correlation kernels: Exponential with power $\mathit{\rho }=2$, Matérn with smoothness $\mathit{\nu }\ge 1$, Rational Quadratic, along with tail rates of the reference prior for these kernels.