Electronic Journal of Statistics

Efficient Gaussian graphical model determination under G-Wishart prior distributions

Hao Wang and Sophia Zhengzi Li

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This paper proposes a new algorithm for Bayesian model determination in Gaussian graphical models under G-Wishart prior distributions. We first review recent development in sampling from G-Wishart distributions for given graphs, with a particular interest in the efficiency of the block Gibbs samplers and other competing methods. We generalize the maximum clique block Gibbs samplers to a class of flexible block Gibbs samplers and prove its convergence. This class of block Gibbs samplers substantially outperforms its competitors along a variety of dimensions. We next develop the theory and computational details of a novel Markov chain Monte Carlo sampling scheme for Gaussian graphical model determination. Our method relies on the partial analytic structure of G-Wishart distributions integrated with the exchange algorithm. Unlike existing methods, the new method requires neither proposal tuning nor evaluation of normalizing constants of G-Wishart distributions.

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Electron. J. Statist., Volume 6 (2012), 168-198.

First available in Project Euclid: 3 February 2012

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Zentralblatt MATH identifier

Exchange algorithms Gaussian graphical models G-Wishart hyper-inverse Wishart Gibbs sampler non-decomposable graphs partial analytic structure posterior simulation


Wang, Hao; Li, Sophia Zhengzi. Efficient Gaussian graphical model determination under G -Wishart prior distributions. Electron. J. Statist. 6 (2012), 168--198. doi:10.1214/12-EJS669. https://projecteuclid.org/euclid.ejs/1328280902

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  • [1] Atay-Kayis, A. and Massam, H. (2005). The marginal likelihood for decomposable and non-decomposable graphical Gaussian models., Biometrika 92 317-35.
  • [2] Bron, C. and Kerbosch, J. (1973). Algorithm 457: finding all cliques of an undirected graph., Communications of the ACM 16 575–577.
  • [3] Carvalho, C., Massam, H. and West, M. (2007). Simulation of hyper-inverse Wishart distributions in graphical models., Biometrika 94 647-659.
  • [4] Carvalho, C. M. and West, M. (2007). Dynamic matrix-variate graphical models., Bayesian Analysis 2 69-98.
  • [5] Dobra, A. and Lenkoski, A. (2011). Copula Gaussian Graphical Models., Annals of Applied Statistics 5 969-993.
  • [6] Dobra, A., Lenkoski, A. and Rodriguez, A. (2011). Bayesian inference for general Gaussian graphical models with application to multivariate lattice data., Journal of the American Statistical Association (to appear).
  • [7] Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks and bonds., Journal of Financial Economics 33 3-56.
  • [8] George, E. I. and McCulloch, R. E. (1997). Approaches for Bayesian variable selection., Statistica Sinica 7 339–373.
  • [9] Giudici, P. and Green, P. J. (1999). Decomposable graphical Gaussian model determination., Biometrika 86 785-801.
  • [10] Godsill, S. J. (2001). On the Relationship Between Markov chain Monte Carlo Methods for Model Uncertainty., Journal of Computational and Graphical Statistics 10 230-248.
  • [11] Jones, B., Carvalho, C., Dobra, A., Hans, C., Carter, C. and West, M. (2005). Experiments in stochastic computation for high-dimensional graphical models., Statistical Science 20 388-400.
  • [12] Kass, R. E., Carlin, B. P., Gelman, A. and Neal, R. M. (1998). Markov Chain Monte Carlo in Practice: A Roundtable Discussion., The American Statistician 52 93-100.
  • [13] Lenkoski, A. and Dobra, A. (2011). Computational Aspects Related to Inference in Gaussian Graphical Models With the G-Wishart Prior., Journal of Computational and Graphical Statistics 20 140-157.
  • [14] Liang, F. (2010). A double Metropolis-Hastings sampler for spatial models with intractable normalizing constants., Journal of Statistical Computing and Simulation 80 1007-1022.
  • [15] Mitsakakis, N., Massam, H. and Escobar, M. (2010). A Metropolis-Hastings based method for sampling from G-Wishart distribution in Gaussian graphical Models., Electronic Journal of Statistics 5 18-31.
  • [16] Murray, I. (2007). Advances in Markov chain Monte Carlo methods PhD Thesis, Gatsby computational neuroscience unit,University College, London.
  • [17] Murray, I., Ghahramani, Z. and MacKay, D. (2006). MCMC for doubly-intractable distributions. In, (Proceedings) Uncertainty in Artificial Intelligence (R. Dechter and T. Richardson, eds.) 359-366. AUAI Press.
  • [18] Pástor, L. and Stambaugh, R. F. (2002). Mutual fund performance and seemingly unrelated assets., Journal of Financial Economics 63 315-349.
  • [19] Piccioni, M. (2000). Independence Structure of Natural Conjugate Densities to Exponential Families and the Gibbs’ Sampler., Scandinavian Journal of Statistics 27 111-127.
  • [20] Rajaratnam, B., Massam, H. and Carvalho, C. M. (2008). Flexible Covariance Estimation in Graphical Gaussian Models., Annals of Statistics 36 2818–49.
  • [21] Robert, C. and Casella, G. (2010)., Monte Carlo Statistical Methods, 2 ed. Springer-Verlag, New York.
  • [22] Rodriguez, A., Lenkoski, A. and Dobra, A. (2011). Sparse covariance estimation in heterogeneous samples., Electronic Journal of Statistics (forthcoming).
  • [23] Roverato, A. (2002). Hyper-Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models., Scandinavian Journal of Statistics 29 391-411.
  • [24] Scott, J. G. and Carvalho, C. M. (2008). Feature-Inclusion Stochastic Search for Gaussian Graphical Models., Journal of Computational and Graphical Statistics 17 790-808.
  • [25] Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk., Journal of Finance 19 425–442.
  • [26] van Dyk, D. A. and Park, T. (2008). Partially Collapsed Gibbs Samplers., Journal of the American Statistical Association 103 790-796.
  • [27] Wang, H. (2010). Sparse seemingly unrelated regression modelling: Applications in finance and econometrics., Computational Statistics & Data Analysis 54 2866-2877.
  • [28] Wang, H. and Carvalho, C. M. (2010). Simulation of hyper-inverse Wishart distributions for non-decomposable graphs., Electronic Journal of Statistics 4 1470-1475.
  • [29] Wang, H., Reeson, C. and Carvalho, C. M. (2011). Dynamic Financial Index Models: Modeling Conditional Dependencies via Graphs., Bayesian Analysis 6 639-664.
  • [30] Wang, H. and West, M. (2009). Bayesian analysis of matrix normal graphical models., Biometrika 96 821-834.