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Bayesian structure learning in graphical models


  • Banerjee, Sayantan
  • Ghosal, Subhashis


We consider the problem of estimating a sparse precision matrix of a multivariate Gaussian distribution, where the dimension p may be large. Gaussian graphical models provide an important tool in describing conditional independence through presence or absence of edges in the underlying graph. A popular non-Bayesian method of estimating a graphical structure is given by the graphical lasso. In this paper, we consider a Bayesian approach to the problem. We use priors which put a mixture of a point mass at zero and certain absolutely continuous distribution on off-diagonal elements of the precision matrix. Hence the resulting posterior distribution can be used for graphical structure learning. The posterior convergence rate of the precision matrix is obtained and is shown to match the oracle rate. The posterior distribution on the model space is extremely cumbersome to compute using the commonly used reversible jump Markov chain Monte Carlo methods. However, the posterior mode in each graph can be easily identified as the graphical lasso restricted to each model. We propose a fast computational method for approximating the posterior probabilities of various graphs using the Laplace approximation approach by expanding the posterior density around the posterior mode. We also provide estimates of the accuracy in the approximation.

Suggested Citation

  • Banerjee, Sayantan & Ghosal, Subhashis, 2015. "Bayesian structure learning in graphical models," Journal of Multivariate Analysis, Elsevier, vol. 136(C), pages 147-162.
  • Handle: RePEc:eee:jmvana:v:136:y:2015:i:c:p:147-162
    DOI: 10.1016/j.jmva.2015.01.015

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    References listed on IDEAS

    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Park, Trevor & Casella, George, 2008. "The Bayesian Lasso," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 681-686, June.
    3. 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.
    4. Jian Guo & Elizaveta Levina & George Michailidis & Ji Zhu, 2011. "Joint estimation of multiple graphical models," Biometrika, Biometrika Trust, vol. 98(1), pages 1-15.
    5. Ghosal, Subhashis, 2000. "Asymptotic Normality of Posterior Distributions for Exponential Families when the Number of Parameters Tends to Infinity," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 49-68, July.
    6. Lam, Clifford & Fan, Jianqing, 2009. "Sparsistency and rates of convergence in large covariance matrix estimation," LSE Research Online Documents on Economics 31540, London School of Economics and Political Science, LSE Library.
    7. McKay Curtis, S. & Banerjee, Sayantan & Ghosal, Subhashis, 2014. "Fast Bayesian model assessment for nonparametric additive regression," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 347-358.
    8. Rothman, Adam J. & Levina, Elizaveta & Zhu, Ji, 2009. "Generalized Thresholding of Large Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 177-186.
    9. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
    10. Yuan, Ming & Lin, Yi, 2005. "Efficient Empirical Bayes Variable Selection and Estimation in Linear Models," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1215-1225, December.
    11. Dobra, Adrian & Hans, Chris & Jones, Beatrix & Nevins, J.R.Joseph R. & Yao, Guang & West, Mike, 2004. "Sparse graphical models for exploring gene expression data," Journal of Multivariate Analysis, Elsevier, vol. 90(1), pages 196-212, July.
    12. Jianhua Z. Huang & Naiping Liu & Mohsen Pourahmadi & Linxu Liu, 2006. "Covariance matrix selection and estimation via penalised normal likelihood," Biometrika, Biometrika Trust, vol. 93(1), pages 85-98, March.
    13. Cai, Tony & Liu, Weidong & Luo, Xi, 2011. "A Constrained â„“1 Minimization Approach to Sparse Precision Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 594-607.
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    Cited by:

    1. Nogales, Francisco J. & Alonso, Andrés M. & Avagyan, Vahe, 2015. "D-trace Precision Matrix Estimation Using Adaptive Lasso Penalties," DES - Working Papers. Statistics and Econometrics. WS 21775, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. repec:eee:jmvana:v:173:y:2019:i:c:p:656-671 is not listed on IDEAS
    3. repec:spr:advdac:v:12:y:2018:i:2:d:10.1007_s11634-016-0272-8 is not listed on IDEAS


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