Vecchia factoring of GP likelihoods

Ignore some conditioning in the dependencies and attain a sparse cholesky factor for the precision matrix

April 27, 2022 — April 27, 2022

graphical models
Hilbert space
kernel tricks
machine learning
Figure 1

There are many ways to cleverly slice up GP likelihoods so that inference is cheap. One is the Vecchia approximation: Approximate the precision matrix by one with a sparse cholesky factorisation.

1 References

Banerjee, Gelfand, Finley, et al. 2008. Gaussian Predictive Process Models for Large Spatial Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Datta, Banerjee, Finley, et al. 2016. Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets.” Journal of the American Statistical Association.
Gramacy. 2016. laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes in R.” Journal of Statistical Software.
Gramacy, and Apley. 2015. Local Gaussian Process Approximation for Large Computer Experiments.” Journal of Computational and Graphical Statistics.
Guinness. 2019. Gaussian Process Learning via Fisher Scoring of Vecchia’s Approximation.” arXiv:1905.08374 [Stat].
Jimenez, and Katzfuss. 2023. Scalable Bayesian Optimization Using Vecchia Approximations of Gaussian Processes.” In Proceedings of The 26th International Conference on Artificial Intelligence and Statistics.
Katzfuss, Guinness, and Lawrence. 2022. Scaled Vecchia Approximation for Fast Computer-Model Emulation.” SIAM/ASA Journal on Uncertainty Quantification.
Khare, and Rajaratnam. 2011. Wishart Distributions for Decomposable Covariance Graph Models.” The Annals of Statistics.
Pardo-Igúzquiza. 1998. Maximum Likelihood Estimation of Spatial Covariance Parameters.” Mathematical Geology.
Peruzzi, Banerjee, and Finley. 2020. Highly Scalable Bayesian Geostatistical Modeling via Meshed Gaussian Processes on Partitioned Domains.” Journal of the American Statistical Association.
Pourahmadi. 2007. Cholesky Decompositions and Estimation of A Covariance Matrix: Orthogonality of Variance–Correlation Parameters.” Biometrika.
Vecchia. 1988. Estimation and Model Identification for Continuous Spatial Processes.” Journal of the Royal Statistical Society: Series B (Methodological).
Zammit-Mangion, Bertolacci, Fisher, et al. 2021. WOMBAT v1.0: A fully Bayesian global flux-inversion framework.” Geoscientific Model Development Discussions.