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Summary For multivariate spatial Gaussian process models, customary specifications of cross-covariance functions do not exploit relational inter-variable graphs to ensure process-level conditional independence between the variables. This is undesirable, especially in highly multivariate settings, where popular cross-covariance functions, such as multivariate Matérn functions, suffer from a curse of dimensionality as the numbers of parameters and floating-point operations scale up in quadratic and cubic order, respectively, with the number of variables. We propose a class of multivariate graphical Gaussian processes using a general construction called stitching that crafts cross-covariance functions from graphs and ensures process-level conditional independence between variables. For the Matérn family of functions, stitching yields a multivariate Gaussian process whose univariate components are Matérn Gaussian processes, and which conforms to process-level conditional independence as specified by the graphical model. For highly multivariate settings and decomposable graphical models, stitching offers massive computational gains and parameter dimension reduction. We demonstrate the utility of the graphical Matérn Gaussian process to jointly model highly multivariate spatial data using simulation examples and an application to air-pollution modelling.
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On perfectness in Gaussian graphical models
Knowing when a graphical model perfectly encodes the conditional independence structure of a distribution is essential in applications, and this is particularly important when performing inference from data. When the model is perfect, there is a one-to-one correspondence between conditional independence statements in the distribution and separation statements in the graph. Previous work has shown that almost all models based on linear directed acyclic graphs as well as Gaussian chain graphs are perfect, the latter of which subsumes Gaussian graphical models (i.e., the undirected Gaussian models) as a special case. In this paper, we directly approach the problem of perfectness for the Gaussian graphical models, and provide a new proof, via a more transparent parametrization, that almost all such models are perfect. Our approach is based on, and substantially extends, a construction of Lněnička and Matúš showing the existence of a perfect Gaussian distribution for any graph. The analysis involves constructing a probability measure on the set of normalized covariance matrices Markov with respect to a graph that may be of independent interest.
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- Award ID(s):
- 1945667
- PAR ID:
- 10336066
- Editor(s):
- Camps-Valls, G.; Ruiz, F. J.; Valera, I.
- Date Published:
- Journal Name:
- Proceedings of The 25th International Conference on Artificial Intelligence and Statistics
- Volume:
- 151
- Page Range / eLocation ID:
- 7505-7517
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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