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August 2021 Propriety of the reference posterior distribution in Gaussian process modeling
Joseph Muré
Author Affiliations +
Ann. Statist. 49(4): 2356-2377 (August 2021). DOI: 10.1214/20-AOS2040


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 ρ<2, Matérn with smoothness ν<1. This paper provides a proof for smooth correlation kernels: Exponential with power ρ=2, Matérn with smoothness ν1, Rational Quadratic, along with tail rates of the reference prior for these kernels.

Funding Statement

The author acknowledges the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-13-MONU-0005 (project CHORUS).


The author would like to thank his PhD advisor, Professor Josselin Garnier (École Polytechnique, Centre de Mathématiques Appliquées), for his guidance, Loic Le Gratiet (EDF R&D, Chatou) and Anne Dutfoy (EDF R&D, Saclay) for their advice and helpful suggestions. He also thanks the Editor, the Associate Editor and the referees for their comments which substantially improved this article.


Download Citation

Joseph Muré. "Propriety of the reference posterior distribution in Gaussian process modeling." Ann. Statist. 49 (4) 2356 - 2377, August 2021.


Received: 1 September 2019; Revised: 1 November 2020; Published: August 2021
First available in Project Euclid: 29 September 2021

Digital Object Identifier: 10.1214/20-AOS2040

Primary: 62F15
Secondary: 60G15 , 62M30

Keywords: correlation function , Gaussian process , integrated likelihood , Jeffreys prior , kriging , posterior propriety , reference prior

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 4 • August 2021
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