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A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models

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  • Aliye Atay-Kayis
  • Helène Massam

Abstract

A centred Gaussian model that is Markov with respect to an undirected graph G is characterised by the parameter set of its precision matrices which is the cone M-super-+(G) of positive definite matrices with entries corresponding to the missing edges of G constrained to be equal to zero. In a Bayesian framework, the conjugate family for the precision parameter is the distribution with Wishart density with respect to the Lebesgue measure restricted to M-super-+(G). We call this distribution the G-Wishart. When G is nondecomposable, the normalising constant of the G-Wishart cannot be computed in closed form. In this paper, we give a simple Monte Carlo method for computing this normalising constant. The main feature of our method is that the sampling distribution is exact and consists of a product of independent univariate standard normal and chi-squared distributions that can be read off the graph G. Computing this normalising constant is necessary for obtaining the posterior distribution of G or the marginal likelihood of the corresponding graphical Gaussian model. Our method also gives a way of sampling from the posterior distribution of the precision matrix. Copyright 2005, Oxford University Press.

Suggested Citation

  • Aliye Atay-Kayis & Helène Massam, 2005. "A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models," Biometrika, Biometrika Trust, vol. 92(2), pages 317-335, June.
    • Handle: RePEc:oup:biomet:v:92:y:2005:i:2:p:317-335
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    File URL: http://hdl.handle.net/10.1093/biomet/92.2.317
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    Cited by:

    1. Atchadé, Yves F., 2019. "Quasi-Bayesian estimation of large Gaussian graphical models," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 656-671.
    2. Banerjee, Sayantan & Ghosal, Subhashis, 2015. "Bayesian structure learning in graphical models," Journal of Multivariate Analysis, Elsevier, vol. 136(C), pages 147-162.
    3. Laurenţiu Cătălin Hinoveanu & Fabrizio Leisen & Cristiano Villa, 2020. "A loss‐based prior for Gaussian graphical models," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 62(4), pages 444-466, December.
    4. Shiers, Nathaniel & Aston, John A.D. & Smith, Jim Q. & Coleman, John S., 2017. "Gaussian tree constraints applied to acoustic linguistic functional data," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 199-215.
    5. Beatrice Franzolini & Alexandros Beskos & Maria De Iorio & Warrick Poklewski Koziell & Karolina Grzeszkiewicz, 2022. "Change point detection in dynamic Gaussian graphical models: the impact of COVID-19 pandemic on the US stock market," Papers 2208.00952, arXiv.org, revised May 2023.
    6. Fitch, A. Marie & Jones, Beatrix, 2012. "The cost of using decomposable Gaussian graphical models for computational convenience," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2430-2441.
    7. Junghi Kim & Kim‐Anh Do & Min Jin Ha & Christine B. Peterson, 2019. "Bayesian inference of hub nodes across multiple networks," Biometrics, The International Biometric Society, vol. 75(1), pages 172-182, March.

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