Scalable iterative methods for sampling from massive Gaussian random vectors

2013/12/05 by Daniel Simpson, Ian Turner, Simpson, Daniel P. +5 · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Numerical Analysis (math.NA) #Statistical Methods and Bayesian Inference

paper · pdf · doi:10.48550/arxiv.1312.1476

openalex publication_date 2013/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Sampling from Gaussian Markov random fields (GMRFs), that is multivariate Gaussian ran- dom vectors that are parameterised by the inverse of their covariance matrix, is a fundamental problem in computational statistics. In this paper, we show how we can exploit arbitrarily accu- rate approximations to a GMRF to speed up Krylov subspace sampling methods. We also show that these methods can be used when computing the normalising constant of a large multivariate Gaussian distribution, which is needed for both any likelihood-based inference method. The method we derive is also applicable to other structured Gaussian random vectors and, in particu- lar, we show that when the precision matrix is a perturbation of a (block) circulant matrix, it is still possible to derive O(n log n) sampling schemes.

Citations

• Specifying Gaussian Markov Random Fields with Incomplete Orthogonal Factorization using Givens Rotations
• Efficient algorithms for the evaluation of the eigenvalues of (block) banded Toeplitz matrices
• Slice sampling covariance hyperparameters of latent Gaussian models
• On the numerical evaluation of Fredholm determinants
• A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow

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• New Heuristics for Parallel and Scalable Bayesian Optimization

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