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Penalized Methods

The Graphical Horseshoe Estimator for Inverse Covariance Matrices

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Pages 747-757 | Received 20 Jul 2017, Accepted 02 Jan 2019, Published online: 29 Apr 2019

References

  • Banerjee, S., and Ghosal, S. (2014), “Posterior Convergence Rates for Estimating Large Precision Matrices Using Graphical Models,” Electronic Journal of Statistics, 8, 2111–2137. DOI: 10.1214/14-EJS945.
  • Banerjee, S., and Ghosal, S. (2015), “Bayesian Structure Learning in Graphical Models,” Journal of Multivariate Analysis, 136, 147–162. DOI: 10.1016/j.jmva.2015.01.015.
  • Barbieri, M. M., and Berger, J. O. (2004), “Optimal Predictive Model Selection,” The Annals of Statistics, 32, 870–897. DOI: 10.1214/009053604000000238.
  • Barron, A. R. (1988), “The Exponential Convergence of Posterior Probabilities With Implications for Bayes Estimators of Density Functions,” Technical Report, Department of Statistics, University of Illinois, Champaign, IL.
  • Bhadra, A., Datta, J., Li, Y., Polson, N. G., and Willard, B. (2016), “Prediction Risk for Global-Local Shrinkage Regression,” arXiv no. 1605.04796.
  • Bhadra, A., Datta, J., Polson, N. G., and Willard, B. (2017), “The Horseshoe + Estimator of Ultra-Sparse Signals,” Bayesian Analysis, 12, 1105–1131. DOI: 10.1214/16-BA1028.
  • Bhadra, A., and Mallick, B. K. (2013), “Joint High-Dimensional Bayesian Variable and Covariance Selection With an Application to eQTL Analysis,” Biometrics, 69, 447–457. DOI: 10.1111/biom.12021.
  • Bhattacharya, A., Pati, D., Pillai, N. S., and Dunson, D. B. (2015), “Dirichlet–Laplace Priors for Optimal Shrinkage,” Journal of the American Statistical Association, 110, 1479–1490. DOI: 10.1080/01621459.2014.960967.
  • Carvalho, C. M., Polson, N. G., and Scott, J. G. (2009), “Handling Sparsity via the Horseshoe,” in International Conference on Artificial Intelligence and Statistics, pp. 73–80.
  • Carvalho, C. M., Polson, N. G., and Scott, J. G. (2010), “The Horseshoe Estimator for Sparse Signals,” Biometrika, 97, 465–480. DOI: 10.1093/biomet/asq017.
  • Cressie, N. (1993), Statistics for Spatial Data, Hoboken, NJ: Wiley.
  • Datta, J., and Ghosh, J. K. (2013), “Asymptotic Properties of Bayes Risk for the Horseshoe Prior,” Bayesian Analysis, 8, 111–132. DOI: 10.1214/13-BA805.
  • Dehmer, M., and Emmert-Streib, F. (2008), Analysis of Microarray Data: A Network-Based Approach, Weinheim, Chichester: Wiley-VCH.
  • Deshpande, S. K., Rockova, V., and George, E. I. (2017), “Simultaneous Variable and Covariance Selection With the Multivariate Spike-and-Slab Lasso,” arXiv no. 1708.08911. DOI: 10.1080/10618600.2019.1593179.
  • Diggle, P. (2002), Analysis of Longitudinal Data, Oxford: Oxford University Press.
  • Fan, J., Feng, Y., and Wu, Y. (2009), “Network Exploration via the Adaptive Lasso and SCAD Penalties,” The Annals of Applied Statistics, 3, 521–541. DOI: 10.1214/08-AOAS215SUPP.
  • Fan, J., and Li, R. (2001), “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties,” Journal of the American Statistical Association, 96, 1348–1360. DOI: 10.1198/016214501753382273.
  • Fan, J., Liao, Y., and Liu, H. (2016), “An Overview of the Estimation of Large Covariance and Precision Matrices,” The Econometrics Journal, 19, C1–C32. DOI: 10.1111/ectj.12061.
  • Friedman, J., Hastie, T., and Tibshirani, R. (2008), “Sparse Inverse Covariance Estimation With the Graphical Lasso,” Biostatistics, 9, 432–441. DOI: 10.1093/biostatistics/kxm045.
  • Friedman, J., Hastie, T., and Tibshirani, R. (2010), “Applications of the Lasso and Grouped Lasso to the Estimation of Sparse Graphical Models,” Technical Report, Stanford University.
  • Friedman, J., Hastie, T., and Tibshirani, R. (2018), “glasso: Graphical Lasso: Estimation of Gaussian Graphical Models,” R Package Version 1.10.
  • Lam, C., and Fan, J. (2009), “Sparsistency and Rates of Convergence in large Covariance Matrix Estimation,” Annals of Statistics, 37, 4254–4278. DOI: 10.1214/09-AOS720.
  • Lee, K., and Lee, J. (2017a), “Estimating Large Precision Matrices via Modified Cholesky Decomposition,” arXiv no. 1707.01143.
  • Lee, K., and Lee, J. (2017b), “Optimal Bayesian Minimax Rates for Unconstrained Large Covariance Matrices,” arXiv no. 1702.07448.
  • Makalic, E., and Schmidt, D. F. (2016), “A Simple Sampler for the Horseshoe Estimator,” IEEE Signal Processing Letters, 23, 179–182. DOI: 10.1109/LSP.2015.2503725.
  • MATLAB (2018), MATLAB, Version R2018a, Natick, MA: The MathWorks Inc.
  • Meinshausen, N., and Bühlmann, P. (2006), “High-Dimensional Graphs and Variable Selection With the Lasso,” The Annals of Statistics, 34, 1436–1462. DOI: 10.1214/009053606000000281.
  • Pati, D., Bhattacharya, A., Pillai, N. S., and Dunson, D. (2014), “Posterior Contraction in Sparse Bayesian Factor Models for Massive Covariance Matrices,” The Annals of Statistics, 42, 1102–1130. DOI: 10.1214/14-AOS1215.
  • Pineda-Pardo, J. A., Bruña, R., Woolrich, M., Marcos, A., Nobre, A. C., Maestú, F., and Vidaurre, D. (2014), “Guiding Functional Connectivity Estimation by Structural Connectivity in MEG: An Application to Discrimination of Conditions of Mild Cognitive Impairment,” Neuroimage, 101, 765–777. DOI: 10.1016/j.neuroimage.2014.08.002.
  • Polson, N. G., and Scott, J. G. (2012), “Local Shrinkage Rules, Lévy Processes and Regularized Regression,” Journal of the Royal Statistical Society, Series B, 74, 287–311. DOI: 10.1111/j.1467-9868.2011.01015.x.
  • Pourahmadi, M. (2011), “Covariance Estimation: The GLM and Regularization Perspectives,” Statistical Science, 26, 369–387. DOI: 10.1214/11-STS358.
  • R Core Team (2018), R: A Language and Environment for Statistical Computing, Vienna, Austria: R Foundation for Statistical Computing.
  • Rajaratnam, B., Massam, H., and Carvalho, C. M. (2008), “Flexible Covariance Estimation in Graphical Gaussian Models,” The Annals of Statistics, 36, 2818–2849. DOI: 10.1214/08-AOS619.
  • Rissanen, J. (1986), “Stochastic Complexity and Modeling,” The Annals of Statistics, 14, 1080–1100. DOI: 10.1214/aos/1176350051.
  • Rothman, A. J., Bickel, P. J., Levina, E., and Zhu, J. (2008), “Sparse Permutation Invariant Covariance Estimation,” Electronic Journal of Statistics, 2, 494–515. DOI: 10.1214/08-EJS176.
  • Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288. DOI: 10.1111/j.2517-6161.1996.tb02080.x.
  • van der Pas, S., Kleijn, B., and van der Vaart, A. (2014), “The Horseshoe Estimator: Posterior Concentration Around Nearly Black Vectors,” Electronic Journal of Statistics, 8, 2585–2618. DOI: 10.1214/14-EJS962.
  • van der Pas, S., Szabó, B., and van der Vaart, A. (2017), “Uncertainty Quantification for the Horseshoe” (with discussion), Bayesian Analysis, 12, 1221–1274. DOI: 10.1214/17-BA1065.
  • Wang, H. (2012), “Bayesian Graphical Lasso Models and Efficient Posterior Computation,” Bayesian Analysis, 7, 867–886. DOI: 10.1214/12-BA729.
  • Xiang, R., Khare, K., and Ghosh, M. (2015), “High Dimensional Posterior Convergence Rates for Decomposable Graphical Models,” Electronic Journal of Statistics, 9, 2828–2854. DOI: 10.1214/15-EJS1084.
  • Yuan, M., and Lin, Y. (2007), “Model Selection and Estimation in the Gaussian Graphical Model,” Biometrika, 94, 19–35. DOI: 10.1093/biomet/asm018.

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