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Efficient Covariance Approximations for Large Sparse Precision Matrices

Journal article
Authors P. Siden
F. Lindgren
David Bolin
M. Villani
Published in Journal of Computational and Graphical Statistics
Volume 27
Issue 4
Pages 898-909
ISSN 1061-8600
Publication year 2018
Published at Department of Mathematical Sciences
Pages 898-909
Language en
Keywords Gaussian Markov random fields, Selected inversion, Sparse precision matrix, Spatial analysis, bayesian-inference, entries, inverse, estimator, models
Subject categories Probability Theory and Statistics


The use of sparse precision (inverse covariance) matrices has become popular because they allow for efficient algorithms for joint inference in high-dimensional models. Many applications require the computation of certain elements of the covariance matrix, such as the marginal variances, which may be nontrivial to obtain when the dimension is large. This article introduces a fast Rao-Blackwellized Monte Carlo sampling-based method for efficiently approximating selected elements of the covariance matrix. The variance and confidence bounds of the approximations can be precisely estimated without additional computational costs. Furthermore, a method that iterates over subdomains is introduced, and is shown to additionally reduce the approximation errors to practically negligible levels in an application on functional magnetic resonance imaging data. Both methods have low memory requirements, which is typically the bottleneck for competing direct methods.

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