Fast matrix algebra for Bayesian model calibration

In Bayesian model calibration, evaluation of the likelihood function usually involves finding the inverse and determinant of a covariance matrix. When Markov Chain Monte Carlo (MCMC) methods are used to sample from the posterior, hundreds of thousands of likelihood evaluations may be required. In th...

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Bibliographic Details
Published in:Journal of statistical computation and simulation 2021-05, Vol.91 (7), p.1331-1341
Main Authors: Rumsey, Kellin N., Huerta, Gabriel
Format: Article
Language:English
Subjects:
41-04
62-04
Bayesian analysis
Bayesian model calibration
Calibration
Covariance matrix
determinant
fast
inverse
likelihood
Markov chains
Mathematical analysis
Matrix algebra
Monte Carlo simulation
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Summary:In Bayesian model calibration, evaluation of the likelihood function usually involves finding the inverse and determinant of a covariance matrix. When Markov Chain Monte Carlo (MCMC) methods are used to sample from the posterior, hundreds of thousands of likelihood evaluations may be required. In this paper, we demonstrate that the structure of the covariance matrix can be exploited, leading to substantial time savings in practice. We also derive two simple equations for approximating the inverse of the covariance matrix in this setting, which can be computed in near-quadratic time. The practical implications of these strategies are demonstrated using a simple numerical case study and the "quack" R package. For a covariance matrix with 1000 rows, application of these strategies for a million likelihood evaluations leads to a speedup of roughly 4000 compared to the naive implementation
ISSN:0094-9655
1563-5163
DOI:10.1080/00949655.2020.1850729