Abstract
We consider a Gaussian variational approximation of the posterior density in high-dimensional state space models. The number of parameters in the covariance matrix of the variational approximation grows as the square of the number of model parameters, so it is necessary to find simple yet effective parametrisations of the covariance structure when the number of model parameters is large. We approximate the joint posterior density of the state vectors by a dynamic factor model, having Markovian time dependence and a factor covariance structure for the states. This gives a reduced description of the dependence structure for the states, as well as a temporal conditional independence structure similar to that in the true posterior. We illustrate the methodology on two examples. The first is a spatio-temporal model for the spread of the Eurasian collared-dove across North America. Our approach compares favorably to a recently proposed ensemble Kalman filter method for approximate inference in high-dimensional hierarchical spatio-temporal models. Our second example is a Wishart-based multivariate stochastic volatility model for financial returns, which is outside the class of models the ensemble Kalman filter method can handle.
Funding Statement
Matias Quiroz and Robert Kohn were partially supported by Australian Research Council Center of Excellence grant CE140100049. David Nott was supported by a Singapore Ministry of Education Academic Research Fund Tier 2 grant (MOE2016-T2-2-135).
Acknowledgments
We thank the Editor, the Associate Editor and two anonymous referees for helping to improve both the content and the presentation of the article. We thank Mevin Hooten for his help with the Eurasian collared-dove data. We thank Linda Tan for her comments on an early version of this manuscript.
Citation
Matias Quiroz. David J. Nott. Robert Kohn. "Gaussian Variational Approximations for High-dimensional State Space Models." Bayesian Anal. 18 (3) 989 - 1016, September 2023. https://doi.org/10.1214/22-BA1332
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