Constructing priors based on model size for nondecomposable Gaussian graphical models: A simulation based approach
A method for constructing priors is proposed that allows the off-diagonal elements of the concentration matrix of Gaussian data to be zero. The priors have the property that the marginal prior distribution of the number of nonzero off-diagonal elements of the concentration matrix (referred to below as model size) can be specified flexibly. The priors have normalizing constants for each model size, rather than for each model, giving a tractable number of normalizing constants that need to be estimated. The article shows how to estimate the normalizing constants using Markov chain Monte Carlo simulation and supersedes the method of Wong et al. (2003)  because it is more accurate and more general. The method is applied to two examples. The first is a mixture of constrained Wisharts. The second is from Wong et al. (2003)  and decomposes the concentration matrix into a function of partial correlations and conditional variances using a mixture distribution on the matrix of partial correlations. The approach detects structural zeros in the concentration matrix and estimates the covariance matrix parsimoniously if the concentration matrix is sparse.
Volume (Year): 102 (2011)
Issue (Month): 5 (May)
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- Frederick Wong, 2003. "Efficient estimation of covariance selection models," Biometrika, Biometrika Trust, vol. 90(4), pages 809-830, December.
- C. M. Carvalho & J. G. Scott, 2009. "Objective Bayesian model selection in Gaussian graphical models," Biometrika, Biometrika Trust, vol. 96(3), pages 497-512.
- Alberto Roverato, 2002. "Hyper Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(3), pages 391-411.
- Smith M. & Kohn R., 2002. "Parsimonious Covariance Matrix Estimation for Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 1141-1153, December.
- John C. Liechty, 2004. "Bayesian correlation estimation," Biometrika, Biometrika Trust, vol. 91(1), pages 1-14, March.
- Daniels, M.J. & Pourahmadi, M., 2009. "Modeling covariance matrices via partial autocorrelations," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2352-2363, November.
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