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Table of contents
Contents
Introduction
I Tools
1 Kriging Overview
1.1 Introduction
1.2 Gaussian random processes
1.3 Mean square continuity and differentiability of Gaussian processes
1.4 Spectral representation
1.5 Examples of correlation kernels
1.6 Current Kriging-related research
2 Reference Prior Theory
2.1 Introduction
2.2 Basic idea
2.3 Full definition of the reference prior
2.4 Regular continuous case
2.5 Properties of reference priors
2.6 Examples
2.7 Reference priors for multiparametric models
3 Propriety of the reference posterior distribution in Gaussian Process regression
3.1 Introduction
3.2 Setting
3.3 Smoothness of the correlation kernel
3.4 Propriety of the reference posterior distribution
3.5 Conclusion
Appendices
3.A Algebraic facts
3.B Maclaurin series
3.C Spectral decomposition
3.D Asymptotic study of the correlation matrix
3.E Details of the proof of Theorem 3.9
II Compromise
4 Optimal compromise between incompatible conditional probability distributions
4.1 Introduction
4.2 Optimal compromise: a general theory
4.3 Discussion of the notion of compromise
4.4 Conclusion
IIIApplication
5 Application of the Optimal Compromise to Simple Kriging models with Matérn correlation kernels
5.1 Introduction
5.2 Optimal compromise between Objective Posterior conditional distributions in Gaussian Process regression
5.3 Comparisons between the MLE and MAP estimators
5.4 Comparison of the predictive distributions associated with the estimators (MLE and MAP) and the full posterior distribution
5.5 Conclusion and Perspectives
5.A Proofs of Section 5.2
6 A Comprehensive Bayesian Treatment of the Universal Kriging model with Matérn correlation kernels
6.1 Introduction
6.2 Analytical treatment of the location-scale parameters
6.3 Reference prior on a one-dimensional
6.4 The Gibbs reference posterior on a multi-dimensional
6.5 Comparison of the predictive performance of the full-Bayesian approach versus MLE and MAP plug-in approaches
6.6 Conclusion
6.A Matérn kernels
6.B Proofs of the existence of the Gibbs reference posterior
7 Trans-Gaussian Kriging in a Bayesian framework: a case study
7.1 Introduction
7.2 Probability Of Detection (POD)
7.3 An Objective Bayesian outlook to Trans-Gaussian Kriging
7.4 Industrial Application
7.5 Conclusion
Conclusion
Bibliography
